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N^kwYI{.Gfɿ-^8ֻGBԺ#,+8֛ӎbJ9gbp.nѳY2;]Zf =8Ian^`ݑL7)acʋ \fDF۪eA+ KP/y;"cNDL/IW>e]l,ij e*=^kJ>SPT ӂ. >d?$+:X@ Lt?&lOfO Q.`8C Z]/;1pKBLV48qniR1Wp %.80+M7KmYஐ WDp~$! bE&s]/?xǧbUޮ6[2OJtm3NuH _7>M]7f zJIPJ'NjC^o2fZBNL)K&;;oh8@RoeAkhD2& bi$O @fժ|ҽ{H~\8Ҋ}vT8ct>GOۂ[N*6 gycV̄Ԙ@Ȟ턏&O۸īL\#f7Td`ꛬ.3bA\em#R֔Yoi.{M U3GެsNJYhn?PBj@:M|@1S|A;0Ejlz=Wu<#gLD߭#VjcC%f&<l~92e-D/ ſxR?a׀@$:YaPM-D]"A|@?l({h@d_(|fP@3`Z/8 !r6%>$mlDڬ.&)j;Lp7Y?Kɍ`M X2ǵ%Mh-hZ@8o=\~ [/=)4y=88lq-2l(h+sޣVTK E#7φ~}iu ~exi_ISҘr| hlo3 N+oN»*&CV,0TZݯǶ<{vur5?m EkMLoUi"Vl26ԕ'*]ĔƔtT@ Yܧq}TJJE!bv7$:P r!]>x {7~x uqni ?H٧{N>uC{FXl!Enn:OWxI@Fb0G`݈鮍NsPGM.lױaSaJ^WGߋGpN*ǙnT9"GiGRWF? }°`jP 6u ~dax\[B,Ϸۿaj.3f?BF0 s -AE}94KGxydմ2B糎:9yd2HHTƷa4>ȧqhrcB>>sZ1_6mM1Il\K]]1Hż|h "YG:& /NQv v-.?G4jJ.l٧ǸRΏܐt]Xăp=H>&UГ(O)h;TA*ꈳٰ(Muj_rcqӲi_Ɠh5b&/ oyvǴVĤO~s {}M`q:Vvk^ߧYRbT#[?8n'0+V@w/DͰ&1Q7+ىu8_qG1V6 \_MOlp@/ʹ͘VcVjξ衟 ؐ.|Ho#^ܕm.uE22)c סmz Y!%2W.!%y;I+V4Mg?ɞKH mI1xvZ+:IH8,4$.dѿl˅xkZyFw[~6A]W ^dX`O~?xd+-n02~c+ $d:z˴ X3ERHd&ʤ74J0';vG]ݏX=qxo*$2t0;΋mkd/sw"FIs9JFї`&K1z ]39nH"IAvK&ZWɲu, }Qx39D`%ƅZ~u FyGR^90'"҂gvs(snE|dՋ7E訝')?0?B#[ʂX[';D6ԺѺBIl i y`-~*1p&šC$K@\"b.uJ7SY@3bcadHDstWSKĪ֪ .;MĊkՂlݐ6|zK-OTycvqOngF'iROFkVHHXLC$;9'܎߸Ek t2)&!W-_݁'mp蛢1#;(M='[IEe:D,ӳU]f(`h= ؟rvCK_Ҁ.2 ̣Xɣwr[ ??paT` C)L362Oi[sshQ::A'@%>~u_4ԑMSGF}EE1ѻQhg0nO9W+6n>B2# q!d<pZY ;܂d T JVYe^SڹHq9j<޳vd:bM6S-|O( 3 \H'V񧄣ͮhǭ~M)u6In @aA𷒍;5d[S II͗ #lb/4CSُ^yMZ?f\Zx9H@`_!~a(%\-=&lޣ6Yr?tN,svykݵpp`2mMs_S69 |}(_2Qi:m\2W?6JM+Yx7{g- ;//( Kdr&r|} c V?06JQG."8Q9m Ή|6\x|7F43":P`H,ݻ?ԺlV"Heb.ߨ}&X<^8L)R ȳ^+>WJSD(r ʞmaH}w4_Yir!Lѥ; ꊓo)5 < .>e(ydgV`Cim ~F Z`󾺝X"/uHD21-a)43Pt".\\=PPxesQ01')tW`$VE]t&Hၙf$vԬNCGEIhOC ,C(I U/a_دt|QFC5*X ~D_Tk"Zĉ{u~9Aɴj~ih[Sfۭ?#mUO\`̥I0<T䎾MU(66Ԁ>vO,SБDuYwxgn*%|2S=*O0 #a;03 ͊.r|& sm:~""b2IvDK~qc&gU|XڟH+hW\[P)1s18QtkLzE57qd\ZJD?&;Ae*}H6V~f>u,9a3%YdGp$+rד{^2f!z9gE1qoA *ٗ!V7m!pm&%.?f ,: 饓]օ3B V/Tc92A4\_# e?u wg'rrV>ڈhpZ~ ػũ-')b-ҷ&yЄqkFGĸ8s<5'QU8oHЍ+a詂{?45Hu,3gUTjXAV99؟ZYgd.jvUK@{3Ӯdj<ȃzE]2IŀȻH%LAN k%&+푡&` $Շ:Og .17߆YȞ@;M(lY< v97ʹ:Q 󊜂 P-I3I+"?;lЩ ,Wp񔬈p)fgD_Mѡ'fV[Y /]@Z/JՓN׬Mw I`2wOz0 *ץٙ!nabyZܒu8FQ]g@n<"@ЖV$[vH!!ZD. 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Cσ endstream endobj 51 0 obj << /Type /Encoding /Differences [ 0 /.notdef 108 /l 109 /.notdef 115 /s 116 /.notdef ] >> endobj 52 0 obj /APWNTH+EUFB10 endobj 53 0 obj << /XHeight 475 /ItalicAngle 0 /Type /FontDescriptor /CapHeight 691 /Flags 4 /StemV 112 /Ascent 691 /FontBBox [ -24 -223 1240 741 ] /Descent -189 /FontName 52 0 R /FontFile 50 0 R >> endobj 54 0 obj [ 336.8 921.3 653.8 609.1 603.7 595.7 459.5 522.7 ] endobj 55 0 obj << /Length1 794 /Length2 1197 /Length3 0 /Length 1991 >> stream %!PS-AdobeFont-1.1: MTSYB 1.000 %%CreationDate: 1996 Aug 26 20:27:19 % Copyright (c) 1996 Michael Spivak. % Hinting Copyright (c) 1996 Y&Y, Inc. 11 dict begin /FontInfo 9 dict dup begin /version (1.000) readonly def /Notice (Copyright (c) 1996 Michael Spivak. Hinting Copyright (c) 1996 Y&Y, Inc.) readonly def /FullName (MTSYB) readonly def /FamilyName (MathTime) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /XTKQHD+MTSYB def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /plus put readonly def /FontBBox{-16 -935 1043 896}readonly def currentdict end currentfile eexec oc;j~EЪxD%L`VBo1++,),%/xPOk2-)Sj{j2B\8_:wǎt%>C0g˙d:VDVE ӕf%t{YHHdiQR+d>d. ɨ2.fFN;ů[zÉj( gŖp-¾!TU/+]&)(KfS%݄0w\: nL& ^j$ң)5>v3]R?mвe/lL@'gCi$owx0x hDWhxVW#H.-˨m 3xKJ7srYh d [GySa9*uڐ|@Q#ԊYK(/-^O=>#lvtu怢qyiJSbaaOTMD\ ێ~MRIۛYW3`g0 {_1}<̥R`|a \#Y@ ur3Qǵ7kvQ&zoZ=#XyQ,\r ubFM_;Y ei%otOD&ĩ>9*G'Ak5#wpo^u;jcg0W,ryɯ"ۅim¨2Ieb)AdQ|[V^n2j> endobj 57 0 obj [ 780 ] endobj 58 0 obj << /Length1 844 /Length2 1861 /Length3 0 /Length 2705 >> stream %!PS-AdobeFont-1.1: RMTMIB 1.000 %%CreationDate: 1996 Aug 26 20:26:39 % Copyright (c) 1996 Michael Spivak. % Hinting Copyright (c) 1996 Y&Y, Inc. 11 dict begin /FontInfo 9 dict dup begin /version (1.000) readonly def /Notice (Copyright (c) 1996 Michael Spivak. Hinting Copyright (c) 1996 Y&Y, Inc.) readonly def /FullName (RMTMIB) readonly def /FamilyName (MathTime) readonly def /Weight (Bold) readonly def /ItalicAngle -14.1 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /HCJFXY+RMTMIB def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 46 /parenleft put dup 47 /parenright put readonly def /FontBBox{0 -208 994 706}readonly def currentdict end currentfile eexec oc;j~EЪ*BgNӽ ؑlKq*޲Xws|QFqv`zXMyp"5O˩YŝP(DT![v67XFlU&3!Rq4wσ~j/Ϥe땨Pq >[98SCν\»tXdnű,)''TAlN/NsYTH_öªo;Iiu#ٗ+kzokWs>A&JOM?#7$RanOe# sYq>-@ (FFy ,Qzwn`MʧN$lRB#>^$NKv_NݳEC*cB"r9?B\`cIUVϭx2,1<:+cut@aGܢ)dJVպ48eB«kU-cGH\l 6u4\Ц~Nx=^VW|LJ plz'`^kyQΆvp' +Ġ}6 D6 7S%Z7˘:/L X6_逋!UoNϡ4y8+B}¤?k+RhI 0ڞud @N+k\ =E؜ʸo>aCwDe]W>˰sbjU0RUUK&2U[,*9p/3Xji3…`IZ:Ø7sնo^SAnI_%Vv$iE*b/Y) @A؀a36=/.: bC}!:70ͻԗ24ؼrY2(gH{0!n+j*Ǯ9% 0xO\5:,W3UMeBArƍ/9Q-C(EF\^1^edÃS6$Ax?XORa_\EvDŃweNמg9Jd7tW;> endobj 60 0 obj [ 376 376 650 804 879 780 832 862 741 772 793 763 331 394 ] endobj 61 0 obj << /Widths 46 0 R /BaseFont /WEEEZL+MTEXB /FirstChar 101 /LastChar 101 /FontDescriptor 45 0 R /Subtype /Type1 /Type /Font >> endobj 62 0 obj << /Widths 49 0 R /BaseFont /EJRITK+t1-gul-bold-italic /FirstChar 108 /LastChar 108 /FontDescriptor 48 0 R /Subtype /Type1 /Type /Font >> endobj 63 0 obj << /Widths 54 0 R /FirstChar 108 /LastChar 115 /FontDescriptor 53 0 R /Subtype /Type1 /Type /Font /BaseFont 52 0 R /Encoding 51 0 R >> endobj 64 0 obj << /Widths 57 0 R /BaseFont /XTKQHD+MTSYB /FirstChar 67 /LastChar 67 /FontDescriptor 56 0 R /Subtype /Type1 /Type /Font >> endobj 65 0 obj << /Widths 60 0 R /BaseFont /HCJFXY+RMTMIB /FirstChar 46 /LastChar 59 /FontDescriptor 59 0 R /Subtype /Type1 /Type /Font >> endobj 66 0 obj << /Length1 1771 /Length2 15290 /Length3 0 /Length 17061 >> stream %!PS-AdobeFont-1.0: t1-uni-regular 002.000 %%Title: t1-uni-regular %Version: 002.000 %%CreationDate: Sat Apr 12 22:19:08 2008 %%Creator: CV Radhakrishnan,,, %Copyright: Copyright (c) 1987, 1991, 1993, 1994, 1998 Adobe Systems %Copyright: Incorporated. All Rights Reserved.Univers is a trademark %Copyright: of Linotype-Hell AG and/or its subsidiaries. % Generated by FontForge 20070607 (http://fontforge.sf.net/) %%EndComments FontDirectory/t1-uni-regular known{/t1-uni-regular findfont dup/UniqueID known{dup /UniqueID get 4172112 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /USCGMP+t1-uni-regular def /FontBBox {-168 -250 992 997 }readonly def /UniqueID 4172112 def /PaintType 0 def /FontInfo 9 dict dup begin /version (002.000) readonly def /Notice (Copyright \050c\051 1987, 1991, 1993, 1994, 1998 Adobe Systems Incorporated. All Rights Reserved.Univers is a trademark of Linotype-Hell AG and/or its subsidiaries.) readonly def /FullName (Univers 55 Roman) readonly def /FamilyName (Univers) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 68 /D put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 58 /colon put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 46 /period put dup 114 /r put dup 115 /s put dup 47 /slash put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put readonly def currentdict end currentfile eexec oc;jAw-ᾉYň5t&נ{hLGqB`d˗Pة|*x\ޖHtEB-gedog7Q`[['W73sZ9 m !tAY!$tY\r[uTIUiBy[Ȝŗ3Kï6zJ!˒Js&B5CV]_I50$zEL>6µ3-O"Ղ,vw ;q8Q^Epd4_PsxO^2|jp5Ժ *Vg/ T"rɵ8t~k W%3M?$ۛWuGt5o6.2|#ԧ@ rد|C1k}')˃Umt8;:?2>GSQs2 KJۋaq%3(jv'T#o映ӲXƴpi-=@ؚi YCuC 纞 Wj/: IqyMG?rKiߣg d@rQKѴ:,] n0a]b g6ˈ}R2,^3>J TŘXbM]&Q2U}o=!X3Sq m0_׬P!C IKV.]!Nu}&QR0NtX.2_%R/d^4^˞Oqq+Q c,ީ)|.'ȣ_Ȁ`( H;1A CF=V˅O׼ s' k܎W;>) aݳaXY7Z bݱϚقf/?8U7>p'p 2ÅHA"&xt՝cs-|2N}i鍜`U+2&cW\LG.51n{uD Q'qGSe9ޡ(!@m:)IΏ;ipU@>\DKCV${Kt \׫̒O̗Ӌ9 ӦL-??sp!/$OMQ':Εz)|Y<^J8\qa_\ތQ*g{@N HnHƌoE kF%'nH깻L}j~RL^lV@vOkc}-e r % ME-eKi7ӈO^+l%"s1ZZ[%)%b??ihogC!4ZuuNT!ؼUpD΅"#pQ]gCǨ}dڌ%=c^C : Tix5j[U?SSӓ]!?dN@pBn.X֚^[AOsDMqAA[ 7?Dv +rʈ7^BS R9M?^ݬ@Q8@b6X.)ՆbgݷDY i$3/̯\! Yll)RQ8xgsaU ApZB:ud*R\ ܶh ?\NIv3!=uEPڇ]Ѐſ֨켃LS/̈-J8jU^4gu7EvE鵈qU :j`-zg?ؿ<*dß-Ͽ;r`_zu{K˗F3\,3*^  }\t׳?_|L'ijGwpXku?K# $}DQ hzO\*8dqS(0c__2F+*"Q鳐JE3NDUA4 ]ֱ21 A0_"j>2qڐcIsA{h/c )Z'ďxrxtr3hͽ Pnd[cwL]#K>O96a6Rzv`+c]# -V(r/.gsIߢh O"!jmk31EvUA8q/(*cf v7GWc&2$ q C"O sy…d{ BpΡ">;Z[!*_=9Gl5GPl,xFcɗ'ZCǸs<~{.#WB#shC ӑA2x>uRi'i?1p;J&k?cV? 4cH!P, @1Št$:gLBWcئzG> (y,F6ծMnQ}IMhcpQ?Bt#ɚyZ8=ٙ/NzKȪ&!FEZTd$Vta3 pW}d{N.\Oj@km=:TnPo-s;5^- VK ˽b#7.2JJw^%\6$@)RQC䕝p*+XK^?FH582N~9VAky΢od=Jc y5[܊$PKI`Lد"Iz=( aodwF=tts(*qYrpyڡ,0z$nA6_ה˞T4KH N{rT(4AIT¯nH93"l(*"TՊz_&Hȱy]k4ԋi R7+L2v:BZ\çkiSd[*#\E鞭!C:#ɌnI$fYu\5M\_ oFMlc0Xi{wŠ%xgNJ!cEtYmě/qey9K!& eڶ]A6(2> wTlPYs.9{D˔aΝJ76ӵbVe2[,%ܘWiPB1I CL5g2c»6V]iU6*F|ID'"?W}6O;l0?j;q"k,u_ĆwD: I 2Sk3}|x$xQMS?ϋ?昝{N4_}euc<,70Gޅ8ۼL&uayΊHa#m93פi̔@NjS6!9#U*%",k/ٿ]M }x piC2ٗ6,Uc+)m w(!hAg*gU-65SHr0G ﷴ$~0`⪯MiF.!Xe0CeEcޣ%]tIt:< Q)Z`*Vs`fntmht\:0җF_:%O$uIqqb'i%>C@x{bFSOd*cZ_ 06krCV^H4 #~bs i8O|GO7:d3Au{CS?)mɴQ;%Ilø_Z/$T~߬[.9zfA3v|'a|a5 J?#YF$+˜Zn%Ԋ<\p 'Wo]VuzȐ1wD딁_i\qzLX&S9$Wx( LIGa=r8vK|,_D$oZNHؑ c |y*Tt]3oB[6>$9q,dylCz /)(S|["$> zQ R/G,YԶ٥X#rP_xOddH$1eE4 l5؛ pQLZU[7"3te{Vװ428&>L;QqI~$,@[b_=\Ve?eeɉ#`CRvNk9\}k! :/BR͑G axe3·d1~T}Ggs=؎1&h*т6N_wo,lX49RU9Suli(=THx" Z^P'!nW߁k~0kGa=1,3ڪG\SXvtQD4j^˺9y(9py@$j)nt),y1g[ʮqt׃:)ŒdLҟ<54 6T\M08(b b{:Hȏ׶\LR\G:fY;2?R%L=qz.ivj֏[C'jSl!Stɶ_ĬMuYTEڰiG:aV<7=(7U/F;&iiسE8.U6SV ƭG'19HrH+Ԍ ۢa3!Xzvk;b(LJܚDD%B2[qg@}<ڭ:He Y9+~?`G0Y&Y-c} =YyX&j %\3^]cMM9b+A YHk2xZTu-lX!x!xy<,򀙄U5d_%B+$1ჷr)\qu-ެnrH/8qbpw=()OzYF1H/$Lt1tLWT\ h*x2 EzJ\hxe:*WT7?}Y ~":ujDQ:oJg9O0M6!6o U9pQ](耝ul!q>yjf:CȢLJG2Y!$ٔ{Ũk{g=+͞IV@H梗1籇5)(l5f1Y4PW!gYA2*(ׄUw}J;AN2ubrTmoOoxv>J 1 -)p>r)$Dϭ@*"{Ht ]pt`TcQ + =ȤQzmpJ0fG@q*͐p~ ??׎VeG1۸Lá9%c.6#؈R}Dui';^-[~dOrGfH"$zGZznRUjQ˼#9qEŽbg[A ief3MWPq  . Z Awpnmp ';p7yC2nnڪdwS6`<ڄ_੝gmˈhɸ Nhυgy=ߞ.9I gH=øY2l"99T;E^ UK rb$&Ƀ ~> 6>ixY`a`y&5Tyfa{TX5 k]FTrISa4 Zmwu??-TO<¶̲#. /=2{( cj=JP8B;W!FN49jSAC:]mU]_Hw,t&}lvSDޕٳRjouH=n{ )ْFX"1{{@ OzdDivrr+GXןAI:5H:)fr}:^fZ5.^՚r>}{~]F@E *k*~?E6Ȓ(䀐)yn9H\"bSg7o,2io` [`mbs]h,񖪹(H=;+׭2-2=.7HITsshfIPb[q}Kf;57;_k)|^Ru{g&vfֺ+ ׎!]!k߉SQum$XEMp@Sf@O/$C`l=>amvFw~p!@i`ـ ԏ g_)j8ߛuNjm#B3~֊ȑM H%jT2 YAPo[@kq9V1m켭pnTw?2QZ*7ܔxg 2]酶 Yt@RpG֥5Mn2Ϙel%| F›B^,D~bXc'tlW=r4d7&Q ˾zR! |~`z$ 3=,T>^/6Z1 %}HKGr G2971ĽBMO~X{=dWe$ ye6p6M@ |qKM\2(P Q#'ĥ_-S{<萦.ZE&Ӝۣ#spď`dW#CM{#,gcH{%MM[RLX_ {ИN!o(12֮2ʕGIW<,zhO[3a7:Gݍ[G{RzJx{ r o\%[T!\^-%/R.ytӘD2}/H%{`WmZ[^L+̩3Y#'$V0m@ђR;]0sʓ`ikЇh _ޒIũՁ}"i2ۚ y\YcnKuaI%|>'_I;OIĂN<x+b{bFf g%,nQnAd:F>ZnRy׏ٜW ;)5"&Di;6Tqzisυjmg2|EG; KycHxpaՅLoZШ`l !?S@~n^t͍vP++jsX[3irļauOf@ (a ԋyKdi* x2|k:N]k8R3 ͏h~G.:ȟ* (w$_!Uwp15t53hcuHv6\ߛj2B~;zcRեIG {'l{& .|HL:Ϳ%FGmJ*5ln0dosRӫbs;iӎ)A0JC[.KX盽-4dEC+lM4aT@l БK5ir{fI|QA[lYS\4Rvxmƥru=ͅ3vPG#뼂^nӉMN[$Ifu;L)1==`q@Fq/!̜QU|E@ѿ'%YR]''"E)rT*\L0yeϊXގG`T0MKkh8esG(|+ebƛt>u+]9ٻ92G5}v J3U_%.YyLF˼fn;qgc儩xt־|۲C lD/R&ٞ!~&|(|M;E8A`J9@em^+i*{0ׯjcǸ8@kx6zcRեIG {$'pLa$+Kб4$˪-xr/V]򩩐-8{2LQ`7G0H5|iJ oͮ ;tT&TE;? 2ҁ_ie.)poϐԥɄo+S{ Yd}[w> =0,?&,H? +Bf% x+2Z <4礐ƒψcH3/U,9S@ondML-Ydr P+KOagU!`&Pc0L5%)պc0X:ў%m0`x{1]HzSZDJ*|9isJנ69W;#8@ ^E~yU3M7`XUbԆSbρzPxQgk]amN1E%4]O<1[޶z\pT:UKb@@-қTU"]}x XqQZ'~ (bmݫ/2T -%\O/D:($2&IC)4[+zF*SSctטtj56Q ai\/#ѯm֯X0϶t07Rv,#HLq 2 TnM#] ONI7j}0n#5n>@-Ki{ݚ@S:(^[t ꕹFMia}lrj5:0O/L>6i9*2hb7g2vx#WjR=3l#̈Tz9-[N|tamb*ZKZUkL}%Gl䩀y'TԚ.&* VeÔ 71]<3)etIOQz|ޗ˭JU=ooxj Հjiy eNa62a2h9 9 Pt`2<#^mF̶a%$s1*ތ= (sD 윗I(dگ´ln!NJy~JYM&n%秇j"cbkM'-3;tpJ}={_EXbgR )*e7`hPVg 43CѲf*ZpR!'2`J瓬v\wFI neL ,A}iC>@qK.%Us=ARQ^7 m'9'jJ F>lm]e֭[rvRfJIDXMs2y ٌ'M{v2oN^)Ri@kg4ɕ5<(느|ŜfU„/\n;d@YRDR-'pZÆ+wmyc>ںObx${`)Ж ]I:RbxcCJ=ZgB'֔qFOD]0\n[\K(Mu _yO"kt̝])HX4I^(1=6QaH:1[F˼3("ȿ6z>x<K|8$>7s.i?uC, \[ۧFX5mȦ_1?RM4MHK7,c.DNԝ1rbM0|q%cI ^G G"ۜ4N (ϝy;iQ[h!9sۃ*neazXm H+d;Y!)\"^*!rUpzO1ܳ8bO`/ zQ\ޥ3xVc PʾC~LČ&j2MEt3bdxs89QӃen(<۪۠a5.wn TbXA;Q?ȋ>g|8'g0~?Jg@!ganRϝUΧ"tcQ D*d/!"W%>Ux&L82j 12#y*i`.RKzY/zFow"t.\N:\&=sDw}r:=j#LD`~؟?BB(Fd@B#Av.<62%S݇:] # /@J<|P=7 l:&J)b2+ -zN ꀑ6^OCtʔQ\Y*ȴ;Jcrnn(l)iW#q_ 7-I`+T0 cEa- 6g:^2 ^AuR6M ʷNES˖VD\KǥK/Y#O .$Qm{ZGC7F?tSj\XA:BrkY!HX@ ~Qt}/=W p5뤅FKCSٵwIT胊 RU+^o8yG nMU[G>#"lc`]z*lny[\?$~wX{uK1>Aဴ$vtaaRbMk3l+O7ĠCƏpM: 6Bm͈L~u##V !@Cus8ht&Rej1CӅHrl`Ŀ,w6i+'> Dë;?f >uxBO7 }pmL܉rݨ9;k0)ks3_3Ȁe)Mf2Qf1EѸ!lÜs12ov!;0*Gʎ}mڐ/{#;ݩZQM6!6lK!@Mx~303 p~t?,+\_F7NkӲ_Ts<*,xz0s䒤qe-9"B p[ӕ!#F'8`(xa :_7'͟1E7[&(˭_qh$$k2 @e,a.l.wQ]'Epm  z`NB!//k?׭`wόť.l> endobj 68 0 obj [ 278 278 556 556 556 556 556 556 556 556 556 556 278 278 584 584 584 500 1015 722 667 667 722 611 556 778 722 278 556 667 556 944 778 778 611 778 667 667 611 778 667 1000 722 667 611 333 278 333 584 500 278 556 611 500 611 556 333 611 611 278 278 500 278 944 611 611 611 611 389 500 333 611 556 833 ] endobj 69 0 obj << /Length1 1369 /Length2 3378 /Length3 0 /Length 4747 >> stream %!PS-AdobeFont-1.1: MTEX 1.1 %%CreationDate: 1993 Jun 26 10:30:36 % Copyright (c) 1992, 1993 The TeXplorators Corporation % Hinting Copyright (c) 1992, 1993 Y&Y, Inc. % MathTime is a trademark of The TeXplorators Corporation 11 dict begin /FontInfo 9 dict dup begin /version (1.1) readonly def /Notice (Copyright (C) 1992, 1993 The TeXplorators Corporation. Hinting (c) 1992, 1993 Y&Y, Inc. (508) 371-3286. All Rights Reserved.) readonly def /FullName (MTEX) readonly def /FamilyName (MathTime) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /ARXZQX+MTEX def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 96 /coproducttext put dup 16 /parenleftBig put dup 32 /parenleftBigg put dup 18 /parenleftbigg put dup 64 /parenleftbt put dup 66 /parenleftex put dup 48 /parenlefttp put dup 17 /parenrightBig put dup 33 /parenrightBigg put dup 19 /parenrightbigg put dup 65 /parenrightbt put dup 67 /parenrightex put dup 49 /parenrighttp put dup 89 /productdisplay put dup 88 /summationdisplay put dup 80 /summationtext put dup 101 /tildewide put dup 91 /uniondisplay put readonly def /FontBBox{-23 -2958 1456 766}readonly def currentdict end currentfile eexec oc;j~EЪ+`U+1;[ihHlgtR@nk{#Y~gJ[. DI~ْ+%hapYz;3NO_O:Vп}KBhIϮԬ";eiwzVcW;I0a1]j"4[{ErGT1QeG0 Vw ߚOҎN'·4b>T(4^mѨnV8Bo8cpWTC 6j'-б^3;ZSi 4LTVC}:)EuF+o1F}NNZ*1aD1U(H`b^;1Mac]&'kM.Lؐ1Sh3\9)Wyϋzoԡ!OWr%\ tA7e:蔧R1ٛ^K3ghʎdFRJJLSq*}:G`>vb+1>5 %N>DoE-\<&NV:Ct\"EzXMQPŞ2Vnte7󛴉'u#-k5R{TD$O#Et =]$5e +'òX@V ME8RUSybc9=Jj.vL'PGs˟ٶRՂu(iKF8vXx>b"7Y ؠe(.З )gJbA㭢[_KMX*E(od^EԶzC~ cw 4l/|bŁT= V?j(E7<ud DG7yDpZC0 c ڢkꋠUQK؇i]fBVepU^Q4PԀ55 ">jGE47]Pc`DpUl߾`n ngS$U\mO0Qd)P3_ nl.p;Z#am"-guxpYtX8!?]럜kRʒC0 O}T^5)2۴-#k;Q_joA]^Lz)aJmJ*o6M۽NH[7T+'~.݋y'a6]MeuuQBceE1 .;?ZQ=XiZ3W{FF՗BBƚ5G'B;dž>} ݄icJ~9nv@誷&ҳ"Z>,A x5baVk:8/Ts- 7c ck?#ѕbd_69Gs,K=I<ˊPg1+ߡ>Gj`Fu,fܴðw6N͟80&~Y;`P.Sld^%30l%~UlZqV$/q Nk72BUx|SnZXѱ󠂃 4 %a CZGaMF]GYlʳ8a%|N&l,3w#Vߜܥ)<9e&P.R3qb=&{y:H yns_pSx95uJ65C6m8v*4MY)yPr܆AfR}\@:  bKnCKMGzuLќbC!}'0]NL q4e~LŞ lyx%M1*]93\2 K}e$X]c[Px#S*Gۅu,wt0L9_4CN2I^7j6ܟ-"iUNCˁQ]Vr&老b|a+Y z`dMاeޮ_5= F ^B`ka JmY L~}?)YX&$۽KO~=T8R jqxfwܮc&2ʬSSwECkiJ=Nʑ!2br?pDm3ڦ{FLq DPwY) ]KeCtKJTFk|"vylж`{Җ M[?kC}HV).LTnMʡSzyWɓv.,zW6'$Z7@lo+-D4Mlѱ 'AE*KU la7i^e^Nd:_ ]Z=BEˍn" endstream endobj 70 0 obj << /XHeight 462 /FontFile 69 0 R /ItalicAngle 0 /Type /FontDescriptor /CapHeight 0 /Flags 4 /StemV 67 /CharSet (/coproducttext/parenleftBig/parenleftBigg/parenleftbigg/parenleftbt/parenleftex/parenlefttp/parenrightBig/parenrightBigg/parenrightbigg/parenrightbt/parenrightex/parenrighttp/productdisplay/summationdisplay/summationtext/tildewide/uniondisplay) /FontName /ARXZQX+MTEX /Ascent 46 /FontBBox [ -23 -2958 1456 766 ] /Descent -600 >> endobj 71 0 obj [ 545 545 675 675 545 545 612 612 612 612 618 618 429 429 1107 1107 693 693 621 621 674 674 674 674 690 690 554 554 1348 1348 866 866 799 799 729 729 729 729 729 729 792 792 792 792 792 792 792 575 799 799 799 799 346 346 984 1235 458 528 1110 1511 1110 1511 1110 1511 1045 940 458 940 940 940 940 940 1415 1269 528 1227 1227 1227 1227 1227 940 1269 742 1075 1408 742 ] endobj 72 0 obj << /Length1 1603 /Length2 4915 /Length3 0 /Length 6518 >> stream %!PS-AdobeFont-1.1: MTSYN 1.000 %%CreationDate: 1996 Aug 26 20:27:17 % Copyright (c) 1996 Michael Spivak. % Hinting Copyright (c) 1996 Y&Y, Inc. 11 dict begin /FontInfo 9 dict dup begin /version (1.000) readonly def /Notice (Copyright (c) 1992-1996 Michael Spivak. Hinting Copyright (c) 1992-1996 Y&Y, Inc.) readonly def /FullName (MTSYN) readonly def /FamilyName (MathTime) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /FontName /RAGOJH+MTSYN def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 104 /angbracketleft put dup 105 /angbracketright put dup 41 /arrowdblright put dup 33 /arrowright put dup 3 /asteriskmath put dup 110 /backslash put dup 106 /bar put dup 102 /braceleft put dup 103 /braceright put dup 84 /bracketleft put dup 85 /bracketright put dup 10 /circlemultiply put dup 8 /circleplus put dup 79 /circumflex put dup 86 /colon put dup 50 /element put dup 59 /emptyset put dup 68 /equal put dup 21 /greaterequal put dup 49 /infinity put dup 92 /intersection put dup 20 /lessequal put dup 0 /minus put dup 7 /minusplus put dup 54 /negationslash put dup 1 /periodcentered put dup 67 /plus put dup 6 /plusminus put dup 48 /prime put dup 26 /propersubset put dup 27 /propersuperset put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 73 /semicolon put dup 81 /tilde put dup 91 /union put readonly def /FontBBox{0 -954 1043 900}readonly def currentdict end currentfile eexec oc;j~EЪڇqcVv3ދF'HJ1ۻ=AWԯ~k=%LT~1%V|St99 KYMyD+,gD}F2Eh#^#dT(@UN0u˰jkV"ozG+1O &`=!ΰ}A/Ve^]=J2j`Kk=eYp6\ Htsc{M|0)luŷ[u;M:= آ]ZS3Ƈ~#y ~qPf= af|#fĬ{ZeOʄݩCs!YՊG IYu~M& 9y|^=C,`Qo aE,rP zs9ε&q[R" sKJ*^#nSC 9Vv"44`Ro?ЬRd(TaJ*)@ivpːyACEs=}|Ng 'J[FG ,]&{!c$1K Wb"cSywGBh9 ա `(͆wTh_7A#y"ᵂ,~d=et78{*|Y: ,wRU}p$f+2܃>H.j{Ms NteikyR śuVb pǓ`cS|cF03RsSn0u)RnC[dk;Ǻ8Z ~s_4 `.Ϣhpȧls=ZZ/\npE_Bc?驾jkp=^\F㓓=, nUe]nCd;V+;e鴀*E)񃬅~bfod A|7XS_TV:.Mf4b$v\jWNў0}0*M@p(,b` Ky1ݙ]3{9q%=4v h=1X-/T:ŹוٜPõMm{ro^ j&YydER>H!7]=J^nC>e!GMU]lPjMP;JE 5ꞩ=V3r.X -@8x bˆwwY> &/:nHs<dqz;{a/tRy(ySΩ#%{> IOArK*siqC➺JD? {[9$뀩Q΍".ߴT;8 ZPbΗ?s}ũSu*WaO(! U肑GrB7t+;z S63{3!i$< Z}UpK29^L;wYYHcӁH9ퟸ<4]ue$Ci\^3?ra:u: [/Yb?\.S'I/ +`A$E{AhCp]P4]jCl&\ >@7`jEWWEAMId;%>6sAVS ɐI*\Zhs.R#(qlq% {@&xrFCdXKLkKu;+pUo2.)DlnD=҄`*m!@Ho0YͧaF~ q j84c,P`l&\>" j0fʬ٨pοF?mתfDZ`kw{ ΔI2sb3;Iް)*bѪ . CEa"ìL*K^nQ *l{45lAbBNs98;-$,InAw61,OKcdyCSx@2*!(k p6۱F޽~_oQɊ4[Zl2u,TWtqX꫊I.UMl;ae |.[Ae[3as1#D wC;U\og`j Norѝ$VbKrb~t]P%Ix,)Y6olF{DWցD=l("l "!~gץtMņ9_PrH皔vl&vpʍRJ.^˒VоIL[1?9LS"%v#(Ey -[ї@C={}oҧOuy^昩L)'H#3S?BL/>¨̞験^H$Ŝif٪"R64Џ|_m>T9^b x [q7V8HP_x ?j *|Bw4"L]߸1 K :ȭRc{4ңxJ !@*'Zb $1\: n}=X깨WOi)-T!xx}!ӭs7G {\DŐsB%<O.NӼbg$,@T J۔c;7⻠bJ6`1o7%ߟ$ Mj>R854M͏ӣ_(w\+t]kT%'8~S~>އ\mӟOlH!d#q-x Sf T1N&d$~@ZeAd6.j՛1Āw^M{~8atNI8;)O%b鈧2h`Lr cjNԐ@!( ^\_OwA,[/_Oc|teʊՆ5] wLB:(zF_93 C "@Yt:u#t˶y(rjU<)kһ_,GKO{ 7'ˀsx8?N^pč%SgO-|nveԠe#Ap%׉.5,r6>-PX`׭L(M-x. IċQ ixK64~̭r>a2~g{CDbJ",¬Ouu}R q VZ[L~W0r5cԐqwK]cZS`nM n>KR#%㘲~j}(oC=ЄS q4*F-9H=8D5~m= *_}A z*#fw-L {Q)Y[bsE[5K>, H5t. ;ߡ[faпUAcܬ6vIj7IͨGX8yZE O ^kI<4y[R7 S,=q(_Nz_ 8fk[j!9_|Uf*(ѡZ( ccoˉX¾)ߙۡ5vICXw4Oyj.k ]oȝ endstream endobj 73 0 obj << /XHeight 460 /FontFile 72 0 R /ItalicAngle 0 /Type /FontDescriptor /CapHeight 362 /Flags 4 /StemV 50 /CharSet (/angbracketleft/angbracketright/arrowdblright/arrowright/asteriskmath/backslash/bar/braceleft/braceright/bracketleft/bracketright/circlemultiply/circleplus/circumflex/colon/element/emptyset/equal/greaterequal/infinity/intersection/lessequal/minus/minusplus/negationslash/periodcentered/plus/plusminus/prime/propersubset/propersuperset/reflexsubset/reflexsuperset/semicolon/tilde/union) /FontName /RAGOJH+MTSYN /Ascent 667 /FontBBox [ 0 -954 1043 900 ] /Descent -90 >> endobj 74 0 obj [ 780 278 784 521 780 556 780 780 800 800 800 800 800 1000 500 500 780 780 780 780 754 754 754 754 780 780 780 780 984 984 754 754 1099 1099 616 616 1043 985 985 780 1043 1043 704 704 1043 985 985 762 270 1021 629 629 784 784 1 1 556 519 722 556 733 635 780 780 634 425 452 780 780 451 536 536 780 357 333 333 333 333 333 333 333 333 347 728 333 333 275 277 700 776 776 668 668 668 700 700 662 662 444 444 444 444 370 370 367 367 286 498 616 711 485 ] endobj 75 0 obj << /Length1 2224 /Length2 6536 /Length3 0 /Length 8760 >> stream %!PS-AdobeFont-1.0: t1-gul-regular-smallcaps 001.001 %%Title: t1-gul-regular-smallcaps %Version: 001.001 %%CreationDate: Mon May 26 10:39:46 2008 %%Creator: CV Radhakrishnan,,, %Copyright: Gulliver, Giant amongst Dwarfs. %Copyright: This typeface offers excellent readability at very small %Copyright: sizes. %Copyright: %Copyright: The digitally encoded machine readable outline data for %Copyright: producing the typefaces licensed to you is copyrighted (c), %Copyright: 1994, 2005 by and the property of Gerard Unger. % Generated by FontForge 20070607 (http://fontforge.sf.net/) %%EndComments FontDirectory/t1-gul-regular-smallcaps known{/t1-gul-regular-smallcaps findfont dup/UniqueID known{dup /UniqueID get 4084430 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 12 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /QDTHNW+t1-gul-regular-smallcaps def /FontBBox {-115 -321 1407 1009 }readonly def /UniqueID 4084430 def /XUID [1021 992 867865856 15320446] def /PaintType 0 def /FontInfo 11 dict dup begin /version (001.001) readonly def /Notice (Gulliver, Giant amongst Dwarfs.\012This typeface offers excellent readability at very small sizes.\012\012The digitally encoded machine readable outline data for producing the typefaces licensed to you is copyrighted \050c\051, 1994, 2005 by and the property of Gerard Unger.) readonly def % Gulliver, Giant amongst Dwarfs. % This typeface offers excellent readability at very small sizes. % % The digitally encoded machine readable outline data for producing the typefaces licensed to you is copyrighted (c), 1994, 2005 by and the property of Gerard Unger. /FullName (Gulliver SCOsF) readonly def /FamilyName (Gulliver SCOsF) readonly def /Weight (Regular) readonly def /FSType 4 def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -75 def /UnderlineThickness 50 def /ascent 789 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put readonly def currentdict end currentfile eexec oc;jAw-36Uu4 l.Nk48DPh>[ME7wk9B.H.CSoZqO O)9OHl~Q:ɾD0j J'JZ `B 5ԆG tbY7:Ϳ2hK@,'$Qr[GB_720 qs`a|Xu\ӅZ?.PcN7.?WֺYq^$^VA7Up c?{RlG9a% #SȣP^-qTo8g^Z#Z #v.Cows?**hfy&;D bÍ!*=8-T9wPa4ԩ^vqB }p ;?IX`D-5 %[!Ӟ1o gn/yhgD`Ce}zB-8Zvʼ-ki8˹HCW*e*~V7gHmWrM)OlPlSN hUGXǧ=v%vӈ ^o66'%&AT( uw7 :L{|&( ,cY44rf*go&z* {NhJl0r1xР%syX\8 ]n`85˲#sBIr:lr܎eǪ7o~!sPAz18xA նNthTw6I[T]`T: v5Z*nZ o)f{H0jer9;Q"}l|a/osARS#݅mrnWe$h1Љ^XY{c+qi|l#mH PJ uSCB`ְ{FflVηiޢɊڿ;%rh0x 1y=XZ |~{rwy)&W$.6K6'ؘǿ6BA%dד\vu;7w}`-!/@ )$9vg޽ȭ2,8tv"# 3 $Buw`f aMG8aOһYbI#mM E.i [qI]4>stU'5upLQvE\ oG~$T2i=$v>ÅN.ȿ&6 ?XxKBal߉X+ıybhrm:dïi9 s܋J`I>:X|xE6l">\#M^>)Ʋ}}s14_;ũ yW-D}#/WML윰k7q7t>O@tlӱ%@WǼD^P ;1 ہZU0jm b{e(zaX ,9s~ƣK ͔?;8C=" C; jc: vhu"w@<]Lجt^}2 ԍnK^j*-FK8:s FBJ27+EP A@F''` H+O 9kI^SL<}Ir>4P :)H1䁜ž5[_RUCAxl=_0 p#ks-K~'G OS@~zk9j?qeVp9rAOAM;n] @`}Bx*QVO9_y|=I_ 8ȅQ({KqKbAanm *.Jh-a~Q&wiܪtP*AH)Hk&E}ZOmzMo\ ּ]8JvxJdo_P lz*o{ۡjWհ#GMN].ɨ2e'%_P2̬W'MPAYQps]_HNs&zt~몙~?K _̤H4GyV(Akj.Q䐧HJ:ux\jLFzoSﴔd!в.- cmnWċַ|Flb`.Tv#HYB\)UґV:ҳ?v?&$AOF GlxbF(qfƂgR]8vWJ7krx7M dyF#qgݸts5r ֤cLm3+NW0: )AI!%A7vac;2FERVނGdE90N!tʆH7l#V]^%2ShԿ CΟ^c.2 ),vo,{(ݞK:LlwM 2Y_?l6EPA>YlQP?7Q%1bt [-F˕<1BoqH D'sRrz,Ռbct"׳1&&3e"G Ƙ# 6|a.T_gCAkm z0Hy-~D"i/3SQEfab6ttMgVMvqZ{."3&Wּ-K$IOɢgz3],tG x֪!K3s(r;h[;pq[^[El2J%HZ9pQAQ7?k11gEbc+ktZE XxKBE{6tNa RSmBD}Dpft d+\|ɃVS+C.5c5 :Mhsė Sc흢ܬJfKMمS*)HÑTۈ~;|ӖJU\5a5=i=ljBDxq!X ̦vJp/Y]D0T Sr;y7}{"NV!![‡M[,jzyhWe lUbڌAz*Bנ/#U/qۏ_ހRVAiR2MO.]LaiN+Uzczi :`vrtWA%FNtQQ:_!؊H ;`C=K-վ3$$qStV,O*;Ky󝳻|l8D0@,e9dvfCS:L7و_X[݇0nc]W3[@gN:hV\LY6vNQXPoRUcc߃'LMs!s'%Rᄟ~lfhQDLn7E P\O%9Tj]|Fe/2͹R tَ@9.-~Rxը /W)Jq4ԓˮbd|)#MV;lY p%9ǖq /s} œ8L6&Pzb G:&=W5F$h1Z~ :txkOk"X%.(AZ(1cH }C}X~cR [`?ձ\c %4ƫ/ 7kct=!Zɸ"/[818!tqxQq}([`$6ɨ߂G*9tH戥>M/&.+MtG1MG(\8f#7p85 MfƭlO8'eq*5ƝDV47L8sZu\ƿ qP~5r{P|)q EDsɺ Yɇ[WSbOH,ƞۏ VF4V Zܰn .ow_吵: %<*Y/sP`6#hlWg6Y i*)fySc[-5GwoK=Ɔ`7%(F`G_\? &A_^7` ڞx᧓1S>%!h< D?AfL̢o}P< 3Zq{BZKAM b&<夔3@s"f_t,בwAˇ-,EK)$6ôOi]KсylZ~ nYNLNbXƻ0Jw,BTHgђSD-\fXC#pZ_&0L:ᓎ0_eW^]ԑŲ]b:LQFWIurV~W=dH#*7Ѷݎ&Z5˫wƮ3o{ GP AzܿBK>&Q]V[:]Fq%ac7xStGp w eꅟ87>d& P(y*{j30ˤ~4G_9&^:]R TQ9&e@as`ij5W=\5< ׃`KD ЗS#| )""9[]?ç!,>(;$+IKr? TR'g>NPl`6CmQ{,%>rtFz=1sQ^TfMpδ7W$WBB#~%jĸ endstream endobj 76 0 obj << /XHeight 566 /FontFile 75 0 R /ItalicAngle 0 /Type /FontDescriptor /CapHeight 742 /Flags 4 /StemV 96 /CharSet (/a/b/c/e/f/i/l/n/o/r/s/t) /FontName /QDTHNW+t1-gul-regular-smallcaps /Ascent 564 /FontBBox [ -115 -321 1407 1009 ] /Descent 0 >> endobj 77 0 obj [ 550 541 528 628 510 476 593 669 316 307 574 455 822 652 611 520 618 562 464 479 ] endobj 78 0 obj << /Length1 3021 /Length2 21250 /Length3 0 /Length 24271 >> stream %!PS-AdobeFont-1.0: t1-gul-regular-italic 001.001 %%Title: t1-gul-regular-italic %Version: 001.001 %%CreationDate: Mon May 26 10:31:00 2008 %%Creator: CV Radhakrishnan,,, %Copyright: Gulliver, Giant amongst Dwarfs. %Copyright: This typeface offers excellent readability at very small %Copyright: sizes. %Copyright: %Copyright: The digitally encoded machine readable outline data for %Copyright: producing the typefaces licensed to you is copyrighted (c), %Copyright: 1994, 2005 by and the property of Gerard Unger. % Generated by FontForge 20070607 (http://fontforge.sf.net/) %%EndComments FontDirectory/t1-gul-regular-italic known{/t1-gul-regular-italic findfont dup/UniqueID known{dup /UniqueID get 4109950 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 12 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /KKGEAU+t1-gul-regular-italic def /FontBBox {-141 -321 1053 1055 }readonly def /UniqueID 4109950 def /XUID [1021 938 1479029120 1353546] def /PaintType 0 def /FontInfo 11 dict dup begin /version (001.001) readonly def /Notice (Gulliver, Giant amongst Dwarfs.\012This typeface offers excellent readability at very small sizes.\012\012The digitally encoded machine readable outline data for producing the typefaces licensed to you is copyrighted \050c\051, 1994, 2005 by and the property of Gerard Unger.) readonly def % Gulliver, Giant amongst Dwarfs. % This typeface offers excellent readability at very small sizes. % % The digitally encoded machine readable outline data for producing the typefaces licensed to you is copyrighted (c), 1994, 2005 by and the property of Gerard Unger. /FullName (Gulliver Italic) readonly def /FamilyName (Gulliver) readonly def /Weight (Regular) readonly def /FSType 4 def /ItalicAngle -8.5 def /isFixedPitch false def /UnderlinePosition -75 def /UnderlineThickness 50 def /ascent 789 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 162 /cacute put dup 58 /colon put dup 44 /comma put dup 100 /d put dup 101 /e put dup 21 /endash put dup 102 /f put dup 53 /five put dup 52 /four put dup 103 /g put dup 104 /h put dup 45 /hyphen put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 57 /nine put dup 111 /o put dup 49 /one put dup 112 /p put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 113 /q put dup 114 /r put dup 115 /s put dup 54 /six put dup 47 /slash put dup 116 /t put dup 51 /three put dup 50 /two put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 48 /zero put readonly def currentdict end currentfile eexec oc;jAw-24 WEvQC<3A2kX0k^|:ٴfb,|=sLu2KZ`3ndgRw@),q՜G<&h'. ΍883 ABAԾI$s@F25ڪ,i pK>6K` %E((bsX5QE8:'1#x[ob0S)õa *eTZD<oWNņHXx͓Tt i?Xn~/G?]3IX6-$r|ƌyS AQXM(u0x۫3+1?&cC1@UV*Z .A"]Y yER\kb7P4 K$Qh+63ήo:U.,tk~uP+J ?"&O H0fS2 .?GPO "/]xkJ"JӚ#B 1u!> ׵ Zvb,:S}2U7I=NEq]`њۿeۺ'B[xe%-g9*fP &y Иrc/8>~i8>LFf|B*G_˙VX0 &: ~'ިP |q)̶m&©+ X^݌i6Cz k"ǬL r]ݚSa="p"?+G| !Y " :A%:δ%{:c?n &f(%)oŕ0NnVL@oi'I_ c `3;z8qefJ U#)rE-oɁ4W(7BT `%G÷>dߋq ;:O<#̄ʃʭ҆vEfhK!%ĻtQ VO4jx.'v84s!QtҴ\ƶJ{)[P 3̞*4,s+%V׎MZe%Cx<{ߕ`Fqo~g_oxhO'o=a`JYBdPWOn^qmYkO}8B=K_3K5`:AVHe6k۲nTcrjt,S@P@'nOh:FPN掏y!8mA{,\DC2\N~4 8Ǎ1DB$eVUB+#MQ{M+8زudAխHq : `͜͜ҵx tJH.VS/ΝL e<^tޑ'RU;Y]f'dG1VW֐-}%0[aD07,kꌪe@ޣWA 6oBsb ^\j!ApxKP!p`( R)a4iWʴ(uRX E9R():+~R֗5I)m8kB +깋[/UtWh SS]ep3zv?νbR<l dk2u'[ʨ `aco Wp&Vmy! !b//6i`x'벑7x  <7NqKZpҰoaZ4M%M.%H"Pΐ=FQ."T9TH֦ —" rb]l"jUI[)`^{GeN.w>˕4%i0&1,aN/ᴦFhCەWфopJ?}'#=KV Ph*H?`+hvfD,>W1;Q݈4ưϫO_97]G? 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/FontName /KKGEAU+t1-gul-regular-italic /Ascent 767 /FontBBox [ -141 -321 1053 1055 ] /Descent -211 >> endobj 80 0 obj [ 579 820 0 931 286 289 600 572 565 861 854 1000 405 413 685 490 565 768 227 373 377 401 549 227 408 229 313 553 553 553 553 553 553 553 553 553 553 316 316 549 549 549 504 779 607 575 577 694 539 486 658 715 301 297 616 474 890 700 682 539 682 587 490 518 687 609 981 605 569 528 343 352 343 549 500 217 533 543 450 556 480 300 514 564 286 289 520 274 864 578 533 556 537 397 408 354 578 517 806 491 524 448 408 277 408 549 0 607 607 577 577 694 539 539 658 474 474 474 700 700 0 682 587 587 490 490 490 518 518 687 687 569 528 528 528 598 301 556 483 533 627 450 ] endobj 81 0 obj << /Length1 1281 /Length2 1259 /Length3 0 /Length 2540 >> stream %!PS-AdobeFont-1.0: t1-gul-symbol 001.000 %%Title: t1-gul-symbol %Version: 001.000 %%CreationDate: Mon May 12 14:48:11 2008 %%Creator: CV Radhakrishnan,,, %Copyright: Created by CV Radhakrishnan,,, with FontForge 2.0 %Copyright: (http://fontforge.sf.net) % 2008-5-12: Created. % Generated by FontForge 20070607 (http://fontforge.sf.net/) %%EndComments FontDirectory/t1-gul-symbol known{/t1-gul-symbol findfont dup/UniqueID known{dup /UniqueID get 4083185 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 12 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /HWNAVD+t1-gul-symbol def /FontBBox {5 -213 895 789 }readonly def /UniqueID 4083185 def /XUID [1021 992 867865827 11599217] def /PaintType 0 def /FontInfo 9 dict dup begin /version (001.000) readonly def /Notice (Created by CV Radhakrishnan,,, with FontForge 2.0 \050http://fontforge.sf.net\051) readonly def /FullName (t1-gul-symbol) readonly def /FamilyName (t1) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 39 /copyright put readonly def currentdict end currentfile eexec oc;jAw-6n}H#vݟ~Lڎko:FVvixؕBv0_ dĮ&8`}-AʌOǦ\Q]&8~a0[V2v)}H6p4aDPf0?IGrq㋒nDJ,52 HEh M/Z1hR 3 |/AF>"T6`~yxd<^;CNJ idjITR⏢Ɂw'ƺVTqرޡc%C TQIC:/*nX̅M}1ba) /h6A!7\PG|~zh~8^8s!BMYI:ꓩua!7h*){7sw2zI۳V,DK-yp~ONi8&ǁHkZr!U-UlM2U6cN=h",Η`%-9 5DpHa>-YÜbh@@XfY|M#X ž endstream endobj 82 0 obj << /XHeight 0 /FontFile 81 0 R /ItalicAngle 0 /Type /FontDescriptor /CapHeight 0 /Flags 4 /StemV 54 /CharSet (/copyright) /FontName /HWNAVD+t1-gul-symbol /Ascent 0 /FontBBox [ 5 -213 895 789 ] /Descent 0 >> endobj 83 0 obj [ 790 ] endobj 84 0 obj << /Length1 3303 /Length2 17859 /Length3 0 /Length 21162 >> stream %!PS-AdobeFont-1.0: t1-gul-regular 001.001 %%Title: t1-gul-regular %Version: 001.001 %%CreationDate: Mon May 26 10:28:22 2008 %%Creator: CV Radhakrishnan,,, %Copyright: Gulliver, Giant amongst Dwarfs. %Copyright: This typeface offers excellent readability at very small %Copyright: sizes. %Copyright: %Copyright: The digitally encoded machine readable outline data for %Copyright: producing the typefaces licensed to you is copyrighted (c), %Copyright: 1994, 2005 by and the property of Gerard Unger. % Generated by FontForge 20070607 (http://fontforge.sf.net/) %%EndComments FontDirectory/t1-gul-regular known{/t1-gul-regular findfont dup/UniqueID known{dup /UniqueID get 4179227 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 12 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /FBCAXJ+t1-gul-regular def /FontBBox {-115 -321 1042 1054 }readonly def /UniqueID 4179227 def /XUID [1021 938 1479029120 7154960] def /PaintType 0 def /FontInfo 11 dict dup begin /version (001.001) readonly def /Notice (Gulliver, Giant amongst Dwarfs.\012This typeface offers excellent readability at very small sizes.\012\012The digitally encoded machine readable outline data for producing the typefaces licensed to you is copyrighted \050c\051, 1994, 2005 by and the property of Gerard Unger.) readonly def % Gulliver, Giant amongst Dwarfs. % This typeface offers excellent readability at very small sizes. % % The digitally encoded machine readable outline data for producing the typefaces licensed to you is copyrighted (c), 1994, 2005 by and the property of Gerard Unger. /FullName (Gulliver) readonly def /FamilyName (Gulliver) readonly def /Weight (Regular) readonly def /FSType 4 def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -75 def /UnderlineThickness 50 def /ascent 789 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 228 /adieresis put dup 64 /at put dup 98 /b put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 162 /cacute put dup 58 /colon put dup 44 /comma put dup 100 /d put dup 36 /dollar put dup 101 /e put dup 233 /eacute put dup 56 /eight put dup 22 /emdash put dup 21 /endash put dup 102 /f put dup 53 /five put dup 52 /four put dup 103 /g put dup 104 /h put dup 45 /hyphen put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 57 /nine put dup 111 /o put dup 49 /one put dup 112 /p put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 113 /q put dup 96 /quoteleft put dup 39 /quoteright put dup 114 /r put dup 115 /s put dup 59 /semicolon put dup 55 /seven put dup 54 /six put dup 47 /slash put dup 116 /t put dup 51 /three put dup 50 /two put dup 117 /u put dup 252 /udieresis put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 48 /zero put readonly def currentdict end currentfile eexec oc;jAw-36Uu4 l.Nk48DPh>[ME7wk9B.H.CSoZqO O)9OHl~Q:ɾD0j 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[([)]TJ/F80 6.8862 Tf -0.756 -2.069 Td [(sl)]TJ/F156 7.6513 Tf 4.973 0 Td [(.)]TJ/F21 6.6311 Tf 2.854 0 Td [(3)]TJ/F156 7.6513 Tf 3.839 0 Td [(/)]TJ/F21 6.6311 Tf 2.854 0 Td [(-modules,)-250(J.)-250(Pure)-250(Appl.)-250(Algebra)-250(210)-250(\0502007\051)-250(559\025575.)]TJ ET 1 0 0 1 40.867 431.116 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -9.822 cm 0 g 0 G 1 0 0 1 -40.867 -421.294 cm BT /F21 6.6311 Tf 40.867 421.294 Td [([12])]TJ ET 1 0 0 1 53.333 421.294 cm 0 g 0 G 1 0 0 1 -53.333 -421.294 cm BT /F21 6.6311 Tf 56.82 421.294 Td [(C.)-327(Calinescu,)-326(J.)-327(Lepowsky,)-327(A.)-327(Milas,)-326(Vertex-algebraic)-327(structure)-327(of)-326(the)-327(principal)-327(subspaces)-326(of)-327(certain)]TJ/F108 6.6311 Tf 306.002 0 Td [(A)]TJ/F156 5.3558 Tf 4.025 3.748 Td [(.)]TJ/F21 4.6417 Tf 1.998 0 Td [(1)]TJ/F156 5.3558 Tf 2.688 0 Td [(/)]TJ/F21 4.6417 Tf -4.686 -5.902 Td [(1)]TJ/F21 6.6311 Tf 7.181 2.154 Td [(-modules,)-327(I:)-326(Level)-327(one)-327(case,)-327(Int.)-326(J.)-327(Math.)-327(19)]TJ ET 1 0 0 1 40.867 421.294 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -421.294 cm BT /F21 6.6311 Tf 56.82 413.324 Td [(\0502008\051)-250(71\02592.)]TJ ET 1 0 0 1 40.867 413.324 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -9.822 cm 0 g 0 G 1 0 0 1 -40.867 -403.502 cm BT /F21 6.6311 Tf 40.867 403.502 Td [([13])]TJ ET 1 0 0 1 53.333 403.502 cm 0 g 0 G 1 0 0 1 -53.333 -403.502 cm BT /F21 6.6311 Tf 56.82 403.502 Td [(C.)-305(Calinescu,)-304(J.)-305(Lepowsky,)-304(A.)-305(Milas,)-305(Vertex-algebraic)-304(structure)-305(of)-305(the)-304(principal)-305(subspaces)-304(of)-305(certain)]TJ/F108 6.6311 Tf 303.952 0 Td [(A)]TJ/F156 5.3558 Tf 4.025 3.748 Td [(.)]TJ/F21 4.6417 Tf 1.998 0 Td [(1)]TJ/F156 5.3558 Tf 2.688 0 Td [(/)]TJ/F21 4.6417 Tf -4.686 -5.902 Td [(1)]TJ/F21 6.6311 Tf 7.182 2.154 Td [(-modules,)-305(II:)-304(Higher-level)-305(case,)-304(J.)-305(Pure)-305(Appl.)]TJ ET 1 0 0 1 40.867 403.502 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -403.502 cm BT /F21 6.6311 Tf 56.82 395.532 Td [(Algebra)-250(212)-250(\0508\051)-250(\0502008\051)-250(1928\0251950.)]TJ ET 1 0 0 1 40.867 395.532 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -9.822 cm 0 g 0 G 1 0 0 1 -40.867 -385.71 cm BT /F21 6.6311 Tf 40.867 385.71 Td [([14])]TJ ET 1 0 0 1 53.333 385.71 cm 0 g 0 G 1 0 0 1 -53.333 -385.71 cm BT /F21 6.6311 Tf 56.82 385.71 Td [(M.)-250(Primc,)-250(Vertex)-250(operator)-250(construction)-250(of)-250(standard)-250(modules)-250(for)]TJ/F108 6.6311 Tf 194.92 0 Td [(A)]TJ/F156 5.3558 Tf 4.025 3.748 Td [(.)]TJ/F21 4.6417 Tf 1.998 0 Td [(1)]TJ/F156 5.3558 Tf 2.687 0 Td [(/)]TJ/F108 4.6417 Tf -4.685 -4.901 Td [(n)]TJ/F21 6.6311 Tf 7.181 1.153 Td [(,)-250(Pacific)-250(J.)-250(Math.)-250(162)-250(\0501994\051)-250(143\025187.)]TJ ET 1 0 0 1 40.867 385.71 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -377.74 cm BT /F21 6.6311 Tf 40.867 377.74 Td [([15])]TJ ET 1 0 0 1 53.333 377.74 cm 0 g 0 G 1 0 0 1 -53.333 -377.74 cm BT /F21 6.6311 Tf 56.82 377.74 Td [(M.)-250(Primc,)-250(Basic)-250(Representations)-250(for)-250(classical)-250(affine)-250(Lie)-250(algebras,)-250(J.)-250(Algebra)-250(228)-250(\0502000\051)-250(1\02550.)]TJ ET 1 0 0 1 40.867 377.74 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -369.77 cm BT /F21 6.6311 Tf 40.867 369.77 Td [([16])]TJ ET 1 0 0 1 53.333 369.77 cm 0 g 0 G 1 0 0 1 -53.333 -369.77 cm BT /F21 6.6311 Tf 56.82 369.77 Td [(B.)-176(Feigin,)-176(M.)-176(Jimbo,)-175(S.)-176(Loktev,)-176(T.)-176(Miwa,)-176(E.)-176(Mukhin,)-176(Bosonic)-175(formulas)-176(for)]TJ/F156 7.6513 Tf 209.159 0 Td [(.)]TJ/F108 6.6311 Tf 2.853 0 Td [(k)]TJ/F156 7.6513 Tf 3.449 0 Td [(;)-167(`/)]TJ/F21 6.6311 Tf 10.242 0 Td [(-admissible)-176(partitions.)]TJ ET 1 0 0 1 351.65 369.77 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -351.65 -369.77 cm BT /F21 6.6311 Tf 351.65 369.77 Td [(math.QA/0107054)]TJ ET 1 0 0 1 407.616 369.77 cm 0 g 0 G 1 0 0 1 -407.616 -369.77 cm BT /F21 6.6311 Tf 407.616 369.77 Td [(;)-176(Addendum)-176(to)-176(`Bosonic)-175(formulas)]TJ ET 1 0 0 1 40.867 369.77 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -369.77 cm BT /F21 6.6311 Tf 56.82 361.8 Td [(for)]TJ/F156 7.6513 Tf 10.344 0 Td [(.)]TJ/F108 6.6311 Tf 2.854 0 Td [(k)]TJ/F156 7.6513 Tf 3.448 0 Td [(;)-167(`/)]TJ/F21 6.6311 Tf 10.243 0 Td [(-admissible)-250(partitions'.)]TJ ET 1 0 0 1 155.371 361.8 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -155.371 -361.8 cm BT /F21 6.6311 Tf 155.371 361.8 Td [(math.QA/0112104)]TJ ET 1 0 0 1 211.337 361.8 cm 0 g 0 G 1 0 0 1 -211.337 -361.8 cm BT /F21 6.6311 Tf 211.337 361.8 Td [(.)]TJ ET 1 0 0 1 40.867 361.8 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -353.83 cm BT /F21 6.6311 Tf 40.867 353.83 Td [([17])]TJ ET 1 0 0 1 53.333 353.83 cm 0 g 0 G 1 0 0 1 -53.333 -353.83 cm BT /F21 6.6311 Tf 56.82 353.83 Td [(B.)-316(Feigin,)-317(M.)-316(Jimbo,)-316(T.)-317(Miwa,)-316(E.)-316(Mukhin,)-317(Y.)-316(Takeyama,)-317(Fermionic)-316(formulas)-316(for)]TJ/F156 7.6513 Tf 239.525 0 Td [(.)]TJ/F108 6.6311 Tf 2.854 0 Td [(k)]TJ/F156 7.6513 Tf 3.448 0 Td [(;)]TJ/F21 6.6311 Tf 3.808 0 Td [(3)]TJ/F156 7.6513 Tf 3.839 0 Td [(/)]TJ/F21 6.6311 Tf 2.854 0 Td [(-admissible)-316(configurations,)-317(Publ.)-316(Res.)-316(Inst.)-317(Math.)-316(Sci.)-316(40)-317(\0502004\051)]TJ ET 1 0 0 1 40.867 353.83 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -353.83 cm BT /F21 6.6311 Tf 56.82 345.86 Td [(125\025162.)]TJ ET 1 0 0 1 40.867 345.86 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -337.89 cm BT /F21 6.6311 Tf 40.867 337.89 Td [([18])]TJ ET 1 0 0 1 53.333 337.89 cm 0 g 0 G 1 0 0 1 -53.333 -337.89 cm BT /F21 6.6311 Tf 56.82 337.89 Td [(S.-J.)-276(Kang,)-275(M.)-276(Kashiwara,)-276(K.C.)-276(Misra,)-275(T.)-276(Miwa,)-276(T.)-276(Nakashima,)-275(A.)-276(Nakayashiki,)-276(Affine)-276(crystals)-275(and)-276(vertex)-276(models,)-276(in:)-275(Proceedings)-276(of)-276(the)-275(RIMS)-276(Research)]TJ ET 1 0 0 1 40.867 337.89 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -337.89 cm BT /F21 6.6311 Tf 56.82 329.92 Td [(Project)-250(1991,)-250(Infinite)-250(Analysis,)-250(in:)-250(International)-250(Journal)-250(of)-250(Modern)-250(Physics)-250(A,)-250(vol.)-250(7,)-250(suppl.)-250(1A,)-250(World)-250(Scientific,)-250(Singapore,)-250(1992,)-250(pp.)-250(449\025484.)]TJ ET 1 0 0 1 40.867 329.92 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.971 cm 0 g 0 G 1 0 0 1 -40.867 -321.949 cm BT /F21 6.6311 Tf 40.867 321.949 Td [([19])]TJ ET 1 0 0 1 53.333 321.949 cm 0 g 0 G 1 0 0 1 -53.333 -321.949 cm BT /F21 6.6311 Tf 56.82 321.949 Td [(M.)-250(Primc,)]TJ/F156 7.6513 Tf 30.39 0 Td [(.)]TJ/F108 6.6311 Tf 2.854 0 Td [(k)]TJ/F156 7.6513 Tf 3.448 0 Td [(;)]TJ/F108 6.6311 Tf 3.808 0 Td [(r)]TJ/F156 7.6513 Tf 3.232 0 Td [(/)]TJ/F21 6.6311 Tf 2.854 0 Td [(-admissible)-250(configurations)-250(and)-250(intertwining)-250(operators,)-250(Contemp.)-250(Math.)-250(442)-250(\0502007\051)-250(425\025434.)]TJ ET 1 0 0 1 40.867 321.949 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -313.979 cm BT /F21 6.6311 Tf 40.867 313.979 Td [([20])]TJ ET 1 0 0 1 53.333 313.979 cm 0 g 0 G 1 0 0 1 -53.333 -313.979 cm BT /F21 6.6311 Tf 56.82 313.979 Td [(V.G.)-250(Kac,)-250(Infinite-Dimensional)-250(Lie)-250(Algebras,)-250(3rd)-250(ed.,)-250(Cambridge)-250(University)-250(Press,)-250(Cambridge,)-250(1990.)]TJ ET 1 0 0 1 40.867 313.979 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -306.009 cm BT /F21 6.6311 Tf 40.867 306.009 Td [([21])]TJ ET 1 0 0 1 53.333 306.009 cm 0 g 0 G 1 0 0 1 -53.333 -306.009 cm BT /F21 6.6311 Tf 56.82 306.009 Td [(C.)-250(Dong,)-250(J.)-250(Lepowsky,)-250(Generalized)-250(Vertex)-250(Algebras)-250(and)-250(Relative)-250(Vertex)-250(Operators,)-250(in:)-250(Progress)-250(in)-250(Math.,)-250(vol.)-250(112,)-250(Birkha\374ser,)-250(Boston,)-250(1993.)]TJ ET 1 0 0 1 40.867 306.009 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -298.039 cm BT /F21 6.6311 Tf 40.867 298.039 Td [([22])]TJ ET 1 0 0 1 53.333 298.039 cm 0 g 0 G 1 0 0 1 -53.333 -298.039 cm BT /F21 6.6311 Tf 56.82 298.039 Td [(I.)-250(Frenkel,)-250(Y.-Z.)-250(Huang,)-250(J.)-250(Lepowsky,)-250(On)-250(axiomatic)-250(approaches)-250(to)-250(vertex)-250(operator)-250(algebras)-250(and)-250(modules,)-250(Mem.)-250(Amer.)-250(Math.)-250(Soc.)-250(104)-250(\0501993\051.)]TJ ET 1 0 0 1 40.867 298.039 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -290.069 cm BT /F21 6.6311 Tf 40.867 290.069 Td [([23])]TJ ET 1 0 0 1 53.333 290.069 cm 0 g 0 G 1 0 0 1 -53.333 -290.069 cm BT /F21 6.6311 Tf 56.82 290.069 Td [(I.)-250(Frenkel,)-250(J.)-250(Lepowsky,)-250(A.)-250(Meurman,)-250(Vertex)-250(Operator)-250(Algebras)-250(and)-250(the)-250(Monster,)-250(in:)-250(Pure)-250(and)-250(Appl.)-250(Math.,)-250(vol.)-250(134,)-250(Academic)-250(Press,)-250(Boston,)-250(1988.)]TJ ET 1 0 0 1 40.867 290.069 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -282.099 cm BT /F21 6.6311 Tf 40.867 282.099 Td [([24])]TJ ET 1 0 0 1 53.333 282.099 cm 0 g 0 G 1 0 0 1 -53.333 -282.099 cm BT /F21 6.6311 Tf 56.82 282.099 Td [(Y.)-250(Gao,)-250(H.-S.)-250(Li,)-250(Generalized)-250(vertex)-250(algebras)-250(generated)-250(by)-250(parafermion-like)-250(vertex)-250(operators,)-250(J.)-250(Algebra)-250(240)-250(\0502001\051)-250(771\025807.)]TJ ET 1 0 0 1 40.867 282.099 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -274.129 cm BT /F21 6.6311 Tf 40.867 274.129 Td [([25])]TJ ET 1 0 0 1 53.333 274.129 cm 0 g 0 G 1 0 0 1 -53.333 -274.129 cm BT /F21 6.6311 Tf 56.82 274.129 Td [(H.-S.)-250(Li,)-250(Local)-250(systems)-250(of)-250(vertex)-250(operators,)-250(vertex)-250(superalgebras)-250(and)-250(modules,)-250(J.)-250(Pure)-250(Appl.)-250(Algebra)-250(109)-250(\0501996\051)-250(143\025195.)]TJ ET 1 0 0 1 422.576 274.129 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -422.576 -274.129 cm BT /F21 6.6311 Tf 422.576 274.129 Td [(hep-th/9406185)]TJ ET 1 0 0 1 472.474 274.129 cm 0 g 0 G 1 0 0 1 -472.474 -274.129 cm BT /F21 6.6311 Tf 472.474 274.129 Td [(.)]TJ ET 1 0 0 1 40.867 274.129 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -266.159 cm BT /F21 6.6311 Tf 40.867 266.159 Td [([26])]TJ ET 1 0 0 1 53.333 266.159 cm 0 g 0 G 1 0 0 1 -53.333 -266.159 cm BT /F21 6.6311 Tf 56.82 266.159 Td [(J.)-250(Lepowsky,)-250(H.-S.)-250(Li,)-250(Introduction)-250(to)-250(Vertex)-250(Operator)-250(Algebras)-250(and)-250(Their)-250(Representations,)-250(in:)-250(Progress)-250(in)-250(Math.,)-250(vol.)-250(227,)-250(Birkh\344user,)-250(Boston,)-250(2004.)]TJ ET 1 0 0 1 40.867 266.159 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -7.97 cm 0 g 0 G 1 0 0 1 -40.867 -258.189 cm BT /F21 6.6311 Tf 40.867 258.189 Td [([27])]TJ ET 1 0 0 1 53.333 258.189 cm 0 g 0 G 1 0 0 1 -53.333 -258.189 cm BT /F21 6.6311 Tf 56.82 258.189 Td [(I.)-250(Frenkel,)-250(V.)-250(Kac,)-250(Basic)-250(representations)-250(of)-250(affine)-250(Lie)-250(algebras)-250(and)-250(dual)-250(resonance)-250(models,)-250(Invent.)-250(Math.)-250(62)-250(\0501980\051)-250(23\02566.)]TJ ET 1 0 0 1 40.867 258.189 cm 0 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[(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(.)-250(But,)-251(the)-250(last)-250(remaining)-251(summand)-250(on)-251(the)-250(right-hand)-250(side)]TJ ET 1 0 0 1 37.134 637.534 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -637.534 cm BT /F21 8.2889 Tf 37.134 627.074 Td [(of)]TJ ET 1 0 0 1 46.949 627.074 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -46.949 -627.074 cm BT /F21 8.2889 Tf 46.949 627.074 Td [(\0506.6\051)]TJ ET 1 0 0 1 64.62 627.074 cm 0 g 0 G 1 0 0 1 -64.62 -627.074 cm BT /F21 8.2889 Tf 66.975 627.074 Td [(is)-284(exactly)-284(the)-284(one)-284(``produced'')-284(by)-284(this)-284(``missing'')-285(tuple.)-284(It)-284(only)-284(remains)-284(to)-284(show)-284(that)-284(each)]TJ/F123 8.2889 Tf 339.383 0 Td [(a)]TJ/F158 9.5641 Tf 7.923 0 Td [(2)]TJ/F193 8.9265 Tf 9.397 0 Td [(B)]TJ/F21 8.2889 Tf 10.246 0 Td [(appears)-284(only)-284(once)]TJ ET 1 0 0 1 37.134 627.074 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -627.074 cm BT /F21 8.2889 Tf 37.134 616.613 Td [(on)-252(the)-253(right-hand)-252(side)-253(of)]TJ ET 1 0 0 1 133.902 616.613 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -133.902 -616.613 cm BT /F21 8.2889 Tf 133.902 616.613 Td [(\0506.6\051)]TJ ET 1 0 0 1 151.574 616.613 cm 0 g 0 G 1 0 0 1 -151.574 -616.613 cm BT /F21 8.2889 Tf 151.574 616.613 Td [(.)-253(For)-252(arbitrary)]TJ/F123 8.2889 Tf 54.246 0 Td [(a)]TJ/F158 9.5641 Tf 7.253 0 Td [(2)]TJ/F193 8.9265 Tf 8.725 0 Td [(B)]TJ/F21 8.2889 Tf 9.984 0 Td [(there)-253(exists)]TJ/F108 8.2889 Tf 46.351 0 Td [(I)]TJ/F158 9.5641 Tf 5.881 0 Td [(2)]TJ/F108 8.2889 Tf 8.726 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.66 0 Td [(such)-253(that)]TJ/F193 8.9265 Tf 37.358 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(I)]TJ/F21 8.2889 Tf 4.811 -3.472 Td [(\050cf.)]TJ ET 1 0 0 1 377.772 616.613 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -377.772 -616.613 cm BT /F21 8.2889 Tf 377.772 616.613 Td [(Lemma)-252(5.6)]TJ ET 1 0 0 1 419.568 616.613 cm 0 g 0 G 1 0 0 1 -419.568 -616.613 cm BT /F21 8.2889 Tf 419.568 616.613 Td [(\051.)-253(But)-252(then,)-253(because)-252(of)]TJ ET 1 0 0 1 37.134 616.613 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -12.029 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -37.134 -604.584 cm BT /F21 8.2889 Tf 37.134 604.584 Td [(Lemma)-271(5.3)]TJ ET 1 0 0 1 79.083 604.584 cm 0 g 0 G 1 0 0 1 -79.083 -604.584 cm BT /F21 8.2889 Tf 79.083 604.584 Td [(,)-271(also)]TJ/F123 8.2889 Tf 21.599 0 Td [(a)]TJ/F158 9.5641 Tf 7.643 0 Td [(2)]TJ/F108 8.2889 Tf 9.117 0 Td [(B)]TJ/F108 5.8022 Tf 4.766 -1.435 Td [(J)]TJ/F21 8.2889 Tf 4.992 1.435 Td [(holds)-271(for)-271(all)]TJ/F108 8.2889 Tf 47.675 0 Td [(J)]TJ/F158 9.5641 Tf 6.313 0 Td [(2)]TJ/F108 8.2889 Tf 9.116 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.812 0 Td [(such)-271(that)]TJ/F108 8.2889 Tf 37.663 0 Td [(J)]TJ/F158 9.5641 Tf 6.313 0 Td [(\022)]TJ/F108 8.2889 Tf 10.561 0 Td [(I)]TJ/F21 8.2889 Tf 3.17 0 Td [(.)-271(This)-271(means)-271(that)-271(the)-271(factor)-271(multip)1(lying)]TJ/F108 8.2889 Tf 153.447 0 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.183 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.393 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F21 8.2889 Tf 5.813 0 Td [(in)]TJ ET 1 0 0 1 37.134 604.584 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -16.556 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -37.134 -588.028 cm BT /F21 8.2889 Tf 37.134 588.028 Td [(\0506.6\051)]TJ ET 1 0 0 1 54.806 588.028 cm 0 g 0 G 1 0 0 1 -54.806 -588.028 cm BT /F21 8.2889 Tf 56.878 588.028 Td [(equals)]TJ/F159 9.5641 Tf 26.74 7.173 Td [(P)]TJ/F158 6.6948 Tf 9.994 -2.382 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F108 5.8022 Tf -4.134 -7.484 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 2.693 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.471 Td [(j)]TJ/F158 6.6948 Tf 1.676 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F159 9.5641 Tf 5.452 7.087 Td [(\020)]TJ/F158 6.6948 Tf 6.408 -6.216 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F108 5.8022 Tf -1.948 -7.738 Td [(j)]TJ/F159 9.5641 Tf 5.058 13.954 Td [(\021)]TJ/F21 8.2889 Tf 5.213 -10.558 Td [(.)-250(But,)-250(this)-250(factor)-250(equals)-250(one:)]TJ ET 1 0 0 1 37.134 588.028 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -588.028 cm BT /F21 8.2889 Tf 61.044 558.783 Td [(1)]TJ/F158 9.5641 Tf 6.925 0 Td [(\000)]TJ/F158 6.6948 Tf 13.327 11.473 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F159 9.5641 Tf -7.876 -2.387 Td [(X)]TJ/F108 5.8022 Tf 1.638 -19.543 Td [(j)]TJ/F158 6.6948 Tf 1.676 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 4.997 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F159 9.5641 Tf 5.452 9.478 Td [(\022)]TJ/F158 9.5641 Tf 7.651 -6.724 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(I)]TJ/F158 9.5641 Tf 3.171 0 Td [(j)]TJ/F108 8.2889 Tf -2.783 -13.495 Td [(j)]TJ/F159 9.5641 Tf 6.714 20.219 Td [(\023)]TJ/F158 9.5641 Tf 9.112 -13.428 Td [(D)]TJ/F158 6.6948 Tf 13.859 11.473 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F159 9.5641 Tf -7.876 -2.387 Td [(X)]TJ/F108 5.8022 Tf 1.637 -19.543 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(0)]TJ/F156 9.5641 Tf 4.997 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(j)]TJ/F159 9.5641 Tf 3.769 9.478 Td [(\022)]TJ/F158 9.5641 Tf 7.651 -6.724 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(I)]TJ/F158 9.5641 Tf 3.17 0 Td [(j)]TJ/F108 8.2889 Tf -2.783 -13.495 Td [(j)]TJ/F159 9.5641 Tf 6.714 20.219 Td [(\023)]TJ/F158 9.5641 Tf 9.112 -13.428 Td [(D)]TJ/F158 6.6948 Tf 13.859 11.473 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F159 9.5641 Tf -7.876 -2.387 Td [(X)]TJ/F108 5.8022 Tf 1.637 -19.543 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(0)]TJ/F156 9.5641 Tf 4.997 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(j)]TJ/F21 8.2889 Tf 2.175 -3.95 Td [(1)]TJ/F158 6.6948 Tf 4.799 3.95 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j\000)]TJ/F108 5.8022 Tf 7.137 0 Td [(j)]TJ/F159 9.5641 Tf 3.769 9.478 Td [(\022)]TJ/F158 9.5641 Tf 7.651 -6.724 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(I)]TJ/F158 9.5641 Tf 3.17 0 Td [(j)]TJ/F108 8.2889 Tf -2.783 -13.495 Td [(j)]TJ/F159 9.5641 Tf 6.714 20.219 Td [(\023)]TJ/F158 9.5641 Tf 9.112 -13.428 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(:)]TJ ET 1 0 0 1 37.134 558.783 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -558.783 cm BT /F21 8.2889 Tf 37.134 533.649 Td [(From)]TJ ET 1 0 0 1 59.207 533.649 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -59.207 -533.649 cm BT /F21 8.2889 Tf 59.207 533.649 Td [(\0506.6\051)]TJ ET 1 0 0 1 76.879 533.649 cm 0 g 0 G 1 0 0 1 -76.879 -533.649 cm BT /F21 8.2889 Tf 78.951 533.649 Td [(we)-250(now)-250(get)]TJ ET 1 0 0 1 37.134 533.649 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -533.649 cm BT /F108 8.2889 Tf 61.044 516.59 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.986 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.286 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 4.355 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 5.446 5.135 Td [(X)]TJ/F123 5.8022 Tf 0.309 -19.251 Td [(a)]TJ/F158 6.6948 Tf 3.18 0 Td [(2)]TJ/F193 6.2485 Tf 4.211 0 Td [(B)]TJ/F108 8.2889 Tf 7.427 10.166 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.183 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.393 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/;)]TJ ET 1 0 0 1 37.134 516.59 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -516.59 cm BT /F21 8.2889 Tf 37.134 492.826 Td [(which)-250(completes)-250(the)-250(inductive)-250(proof.)]TJ/F137 7.9701 Tf 154.404 0 Td [(\003)]TJ ET 1 0 0 1 37.134 492.826 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -17.835 cm 0 g 0 G 1 0 0 1 -33.149 -474.991 cm BT /F123 8.2889 Tf 37.134 474.991 Td [(Proposition)-250(6.3.)]TJ ET 1 0 0 1 100.121 474.991 cm 0 g 0 G 1 0 0 1 -100.121 -474.991 cm BT /F108 8.2889 Tf 104.106 474.991 Td [(For)-301(all)-300(choices)-301(of)-301(nonnegative)-300(integers)-301(k)]TJ/F21 5.8022 Tf 147.154 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F108 8.2889 Tf 6.124 1.434 Td [(such)-301(that)-386(k)]TJ/F21 5.8022 Tf 41.543 -1.434 Td [(0)]TJ/F158 9.5641 Tf 6.414 1.434 Td [(C)-267(\001)-167(\001)-167(\001)-267(C)]TJ/F108 8.2889 Tf 33.754 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 9.5641 Tf 7.366 1.434 Td [(D)]TJ/F108 8.2889 Tf 11.195 0 Td [(k)-301(and)-300(all)-301(nonnegative)-301(integers)]TJ ET 1 0 0 1 37.134 474.991 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -474.991 cm BT /F108 8.2889 Tf 37.134 464.53 Td [(n)]TJ/F21 5.8022 Tf 4.791 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(n)]TJ/F156 6.6948 Tf 4.791 -1.434 Td [(`)]TJ/F108 8.2889 Tf 5.704 1.434 Td [(such)-250(that)-335(n)]TJ/F108 5.8022 Tf 41.183 -1.434 Td [(i)]TJ/F158 9.5641 Tf 4.814 1.434 Td [(\025)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.434 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.434 Td [(,)-250(i)]TJ/F158 9.5641 Tf 8.981 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F108 8.2889 Tf 26.668 0 Td [(,)-250(the)-250(following)-250(assertion)-250(is)-250(true:)]TJ ET 1 0 0 1 37.134 464.53 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -464.53 cm BT /F108 8.2889 Tf 61.044 434.038 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.286 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 4.355 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 19.567 6.704 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.472 Td [(n)]TJ ET q 1 0 0 1 115.352 436.429 cm []0 d 0 J 0.44 w 0 0 m 24.813 0 l S Q BT /F21 8.2889 Tf 115.352 427.247 Td [(1)]TJ/F158 9.5641 Tf 6.925 0 Td [(\000)]TJ/F108 8.2889 Tf 9.585 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 2.764 Td [(n)]TJ/F159 9.5641 Tf 6.642 26.063 Td [(0)]TJ 0 -16.776 Td [(B)]TJ 0 -6.178 Td [(@)]TJ 14.958 10.004 Td [(X)]TJ/F123 4.9734 Tf 1.231 -18.657 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(2)]TJ/F193 5.3559 Tf 3.609 0 Td [(B)]TJ/F123 4.9734 Tf -13.687 -5.416 Td [(a)]TJ/F158 5.7385 Tf 2.725 0 Td [(6)1(D)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F21 4.9734 Tf 2.141 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(/)]TJ/F108 8.2889 Tf 4.93 14.987 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F158 6.6948 Tf 3.183 1.392 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.393 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.477 0 Td [(1)]TJ/F156 9.5641 Tf 14.04 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 18.635 5.136 Td [(X)]TJ/F123 4.9734 Tf -11.994 -18.918 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(2)]TJ/F193 5.3559 Tf 3.609 0 Td [(B)]TJ/F108 4.9734 Tf 4.392 -1.705 Td [(k)]TJ/F156 5.7385 Tf 2.586 0 Td [(;)]TJ/F21 4.9734 Tf 1.9 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F158 5.7385 Tf 3.378 1.705 Td [(n)]TJ/F193 5.3559 Tf 2.783 0 Td [(B)]TJ/F123 4.9734 Tf -26.913 -7.386 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(6)1(D)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F21 4.9734 Tf 2.14 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(/)]TJ/F108 8.2889 Tf 10.803 17.218 Td [(A)]TJ/F108 5.8022 Tf 5.032 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.392 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.092 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F159 9.5641 Tf 3.568 22.036 Td [(1)]TJ 0 -16.776 Td [(C)]TJ 0 -6.178 Td [(A)]TJ/F156 9.5641 Tf 9.235 0.918 Td [(:)]TJ ET 1 0 0 1 37.134 434.038 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -35.067 cm 0 g 0 G 1 0 0 1 -33.149 -398.971 cm BT /F123 8.2889 Tf 37.134 398.971 Td [(Proof.)]TJ ET 1 0 0 1 61.362 398.971 cm 0 g 0 G 1 0 0 1 -61.362 -398.971 cm BT /F21 8.2889 Tf 65.347 398.971 Td [(From)]TJ ET 1 0 0 1 87.42 398.971 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -87.42 -398.971 cm BT /F21 8.2889 Tf 87.42 398.971 Td [(\0506.5\051)]TJ ET 1 0 0 1 105.092 398.971 cm 0 g 0 G 1 0 0 1 -105.092 -398.971 cm BT /F21 8.2889 Tf 107.164 398.971 Td [(we)-250(have)]TJ ET 1 0 0 1 37.134 398.971 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -398.971 cm BT /F108 8.2889 Tf 61.044 381.912 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.287 Td [(k)]TJ/F156 6.6948 Tf 3.017 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 9.5641 Tf 5.175 2.693 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 7.054 0 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 14.575 5.136 Td [(X)]TJ/F123 5.8022 Tf -9.129 -19.252 Td [(a)]TJ/F158 6.6948 Tf 3.18 0 Td [(2)]TJ/F193 6.2485 Tf 4.211 0 Td [(B)]TJ/F108 4.9734 Tf 5.124 -1.353 Td [(k)]TJ/F156 5.7385 Tf 2.586 0 Td [(;)]TJ/F21 4.9734 Tf 1.899 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F108 8.2889 Tf 4.972 11.519 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.392 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.392 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.452 0 Td [(/)]TJ ET 1 0 0 1 37.134 381.912 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -381.912 cm BT /F158 9.5641 Tf 104.87 355.246 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 14.876 5.136 Td [(X)]TJ/F123 4.9734 Tf -8.234 -18.657 Td [(a)]TJ/F158 5.7385 Tf 2.725 0 Td [(2)]TJ/F193 5.3559 Tf 3.61 0 Td [(B)]TJ/F108 4.9734 Tf 4.391 -1.705 Td [(k)]TJ/F156 5.7385 Tf 2.587 0 Td [(;)]TJ/F21 4.9734 Tf 1.899 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F123 4.9734 Tf -24.511 -5.681 Td [(a)]TJ/F158 5.7385 Tf 2.725 0 Td [(6)1(D)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F21 4.9734 Tf 2.141 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(/)]TJ/F108 8.2889 Tf 7.044 16.957 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F158 6.6948 Tf 3.183 1.392 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.393 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.477 0 Td [(1)]TJ/F156 9.5641 Tf 14.04 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F108 8.2889 Tf 3.852 -3.95 Td [(A)]TJ/F108 5.8022 Tf 5.031 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.287 Td [(k)]TJ/F156 6.6948 Tf 3.017 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 9.5641 Tf 5.174 2.693 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/;)]TJ ET 1 0 0 1 37.134 355.246 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -355.246 cm BT /F21 8.2889 Tf 37.134 323.673 Td [(and)-250(it)-250(follows)-250(that)]TJ ET 1 0 0 1 37.134 323.673 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -323.673 cm BT /F108 8.2889 Tf 61.044 302.202 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.287 Td [(k)]TJ/F156 6.6948 Tf 3.017 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 9.5641 Tf 5.175 2.693 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 19.567 6.705 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.471 Td [(n)]TJ ET q 1 0 0 1 115.352 304.593 cm []0 d 0 J 0.44 w 0 0 m 24.813 0 l S Q BT /F21 8.2889 Tf 115.352 295.412 Td [(1)]TJ/F158 9.5641 Tf 6.925 0 Td [(\000)]TJ/F108 8.2889 Tf 9.585 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 2.764 Td [(n)]TJ/F159 9.5641 Tf 16.072 13.112 Td [(X)]TJ/F123 4.9734 Tf -8.235 -18.657 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(2)]TJ/F193 5.3559 Tf 3.609 0 Td [(B)]TJ/F108 4.9734 Tf 4.392 -1.705 Td [(k)]TJ/F156 5.7385 Tf 2.586 0 Td [(;)]TJ/F21 4.9734 Tf 1.899 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F123 4.9734 Tf -24.511 -5.681 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(6)1(D)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F21 4.9734 Tf 2.14 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(/)]TJ/F108 8.2889 Tf 7.044 16.957 Td [(A)]TJ/F108 5.8022 Tf 5.032 5.389 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.092 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/:)]TJ/F21 8.2889 Tf 241.454 0 Td [(\0506.7\051)]TJ ET 1 0 0 1 37.134 302.202 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -302.202 cm BT /F21 8.2889 Tf 37.134 270.63 Td [(Let)-250(us)-250(now)-250(fix)-250(nonnegative)-250(integers)]TJ/F108 8.2889 Tf 137.644 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F21 8.2889 Tf 5.704 1.435 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.316 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(k)]TJ/F21 8.2889 Tf 4.311 0 Td [(.)-250(From)]TJ ET 1 0 0 1 346.34 270.63 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -346.34 -270.63 cm BT /F21 8.2889 Tf 346.34 270.63 Td [(\0506.5\051)]TJ ET 1 0 0 1 364.012 270.63 cm 0 g 0 G 1 0 0 1 -364.012 -270.63 cm BT /F21 8.2889 Tf 366.084 270.63 Td [(by)-250(using)]TJ ET 1 0 0 1 400.25 270.63 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -400.25 -270.63 cm BT /F21 8.2889 Tf 400.25 270.63 Td [(\0506.7\051)]TJ ET 1 0 0 1 417.922 270.63 cm 0 g 0 G 1 0 0 1 -417.922 -270.63 cm BT /F21 8.2889 Tf 419.994 270.63 Td [(we)-250(get)]TJ ET 1 0 0 1 37.134 270.63 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -270.63 cm BT /F108 8.2889 Tf 61.044 253.57 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.286 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 9.5641 Tf 4.355 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 7.054 0 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 5.446 5.136 Td [(X)]TJ/F123 5.8022 Tf 0.31 -19.251 Td [(a)]TJ/F158 6.6948 Tf 3.179 0 Td [(2)]TJ/F193 6.2485 Tf 4.211 0 Td [(B)]TJ/F108 8.2889 Tf 7.427 10.165 Td [(A)]TJ/F108 5.8022 Tf 5.032 5.389 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.092 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 37.134 253.57 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -253.57 cm BT /F158 9.5641 Tf 104.87 216.73 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 5.446 18.086 Td [(0)]TJ 0 -16.775 Td [(B)]TJ 0 -6.179 Td [(@)]TJ/F108 8.2889 Tf 7.642 0.918 Td [(A)]TJ/F108 5.8022 Tf 5.031 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.287 Td [(k)]TJ/F156 6.6948 Tf 3.017 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 9.5641 Tf 5.174 2.693 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.692 0 Td [(C)]TJ/F159 9.5641 Tf 16.902 9.086 Td [(X)]TJ/F123 4.9734 Tf 1.232 -18.657 Td [(a)]TJ/F158 5.7385 Tf 2.725 0 Td [(2)]TJ/F193 5.3559 Tf 3.61 0 Td [(B)]TJ/F123 4.9734 Tf -13.688 -5.416 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(6)1(D)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F21 4.9734 Tf 2.14 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(/)]TJ/F108 8.2889 Tf 4.93 14.987 Td [(A)]TJ/F108 5.8022 Tf 5.032 5.389 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.092 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F159 9.5641 Tf 3.568 22.036 Td [(1)]TJ 0 -16.776 Td [(C)]TJ 0 -6.178 Td [(A)]TJ ET 1 0 0 1 37.134 216.73 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -216.73 cm BT /F158 9.5641 Tf 104.87 173.572 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 5.446 18.086 Td [(0)]TJ 0 -16.776 Td [(B)]TJ 0 -6.178 Td [(@)]TJ/F108 8.2889 Tf 17.092 7.623 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.471 Td [(n)]TJ ET q 1 0 0 1 134.551 175.963 cm []0 d 0 J 0.44 w 0 0 m 24.813 0 l S Q BT /F21 8.2889 Tf 134.551 166.782 Td [(1)]TJ/F158 9.5641 Tf 6.925 0 Td [(\000)]TJ/F108 8.2889 Tf 9.585 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 2.764 Td [(n)]TJ/F159 9.5641 Tf 16.072 13.112 Td [(X)]TJ/F123 4.9734 Tf -8.235 -18.657 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(2)]TJ/F193 5.3559 Tf 3.609 0 Td [(B)]TJ/F108 4.9734 Tf 4.392 -1.705 Td [(k)]TJ/F156 5.7385 Tf 2.586 0 Td [(;)]TJ/F21 4.9734 Tf 1.9 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F123 4.9734 Tf -24.511 -5.681 Td [(a)]TJ/F158 5.7385 Tf 2.726 0 Td [(6)1(D)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F21 4.9734 Tf 2.14 0 Td [(0)]TJ/F156 5.7385 Tf 2.88 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.533 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(/)]TJ/F108 8.2889 Tf 7.044 16.957 Td [(A)]TJ/F108 5.8022 Tf 5.032 5.389 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.092 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 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5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 9.5641 Tf 5.47 -2.597 Td [(:)-167(:)-166(:)-167(.)]TJ/F108 8.2889 Tf 16.24 0 Td [(z)]TJ/F156 6.6948 Tf 3.713 -1.435 Td [(`)]TJ/F108 8.2889 Tf 3.631 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.875 -2.557 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.827 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(z)]TJ/F156 6.6948 Tf 3.714 -1.435 Td [(`)]TJ/F108 8.2889 Tf 3.631 1.435 Td [(q)]TJ/F158 9.5641 Tf 4.451 0 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/:)]TJ/F21 8.2889 Tf 93.33 0 Td [(\0506.2\051)]TJ ET 1 0 0 1 40.867 661.438 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -661.438 cm BT /F21 8.2889 Tf 40.867 635.716 Td [(Since)]TJ ET 1 0 0 1 63.989 635.716 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -63.989 -635.716 cm BT /F21 8.2889 Tf 63.989 635.716 Td [(\0506.1\051)]TJ ET 1 0 0 1 81.661 635.716 cm 0 g 0 G 1 0 0 1 -81.661 -635.716 cm BT /F21 8.2889 Tf 84.584 635.716 Td [(holds)-353(for)]TJ/F108 8.2889 Tf 37.534 0 Td [(W)]TJ/F158 9.5641 Tf 14.084 0 Td [(D)]TJ/F108 8.2889 Tf 12.299 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.856 1.435 Td [(C)-313(\001)-167(\001)-167(\001)-313(C)]TJ/F108 8.2889 Tf 35.078 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.631 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/)]TJ/F21 8.2889 Tf 6.491 0 Td [(for)-353(all)-352(possible)-353(choices)-352(of)-353(nonnegative)-353(integers)]TJ/F108 8.2889 Tf 185.665 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F21 8.2889 Tf 6.554 1.435 Td [(such)-353(that)]TJ ET 1 0 0 1 40.867 635.716 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -635.716 cm BT /F108 8.2889 Tf 40.867 625.255 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 4.935 1.435 Td [(C)-113(\001)-166(\001)-167(\001)-113(C)]TJ/F108 8.2889 Tf 29.318 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 8.2889 Tf 4.31 0 Td [(,)-209(we)-209(can)-209(say)-209(that)]TJ ET 1 0 0 1 167.854 625.255 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -167.854 -625.255 cm BT /F21 8.2889 Tf 167.854 625.255 Td [(\0506.1\051)]TJ ET 1 0 0 1 185.526 625.255 cm 0 g 0 G 1 0 0 1 -185.526 -625.255 cm BT /F21 8.2889 Tf 187.257 625.255 Td [(is)-209(a)-209(system)-209(of)-209(relations.)-209(Our)-209(attempt)-209(will)-209(be)-209(directed)-208(towards)-209(proving)-209(the)-209(uniqueness)]TJ ET 1 0 0 1 40.867 625.255 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -625.255 cm BT /F21 8.2889 Tf 40.867 614.794 Td [(of)-250(solution)-250(for)-250(this)-250(system.)]TJ ET 1 0 0 1 40.867 614.794 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -20.338 cm 0 g 0 G 1 0 0 1 -36.882 -594.456 cm BT /F123 8.2889 Tf 40.867 594.456 Td [(Example)-250(6.1.)]TJ ET 1 0 0 1 91.437 594.456 cm 0 g 0 G 1 0 0 1 -91.437 -594.456 cm BT /F108 8.2889 Tf 95.422 594.456 Td [(In)-250(the)-250(case)-250(of)]TJ/F156 9.5641 Tf 51.254 0 Td [(`)]TJ/F158 9.5641 Tf 7.133 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(2)]TJ/F108 8.2889 Tf 4.799 0 Td [(,)-250(k)]TJ/F158 9.5641 Tf 10.921 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(2)]TJ/F108 8.2889 Tf 4.799 0 Td [(,)-250(the)-250(above)-250(system)-250(reads:)]TJ ET 1 0 0 1 40.867 594.456 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -594.456 cm BT /F156 9.5641 Tf 64.777 578.103 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(2)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F156 9.5641 Tf 9.585 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F21 5.8022 Tf 3.567 3.95 Td [(2)]TJ/F156 9.5641 Tf 3.858 -3.95 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(2)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F158 9.5641 Tf 4.452 0 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 40.867 578.103 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -578.103 cm BT /F156 9.5641 Tf 64.777 564.41 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 8.549 0 Td [(D)]TJ/F156 9.5641 Tf 12.441 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F156 9.5641 Tf 9.585 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.134 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.857 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.452 0 Td [(/)]TJ/F158 9.5641 Tf -154.845 -10.461 Td [(\000)]TJ/F156 9.5641 Tf 7.46 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.434 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.434 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F156 9.5641 Tf 9.585 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.434 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(2)]TJ/F108 8.2889 Tf 3.858 1.434 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/\037)-100(.)]TJ/F108 8.2889 Tf 13.715 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.434 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.434 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.434 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(2)]TJ/F108 8.2889 Tf 3.858 1.434 Td [(q)]TJ/F158 9.5641 Tf 4.451 0 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.452 0 Td [(/)]TJ ET 1 0 0 1 40.867 553.949 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -553.949 cm BT /F156 9.5641 Tf 64.777 540.5 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.215 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F158 9.5641 Tf 4.451 0 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 40.867 540.5 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -540.5 cm BT /F156 9.5641 Tf 64.777 525.727 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F156 9.5641 Tf 9.585 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F21 5.8022 Tf 3.567 3.95 Td [(2)]TJ/F156 9.5641 Tf 3.858 -3.95 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(2)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F158 9.5641 Tf 4.452 0 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 40.867 525.727 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -525.727 cm BT /F156 9.5641 Tf 64.777 512.277 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(2)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.215 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(2)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F158 9.5641 Tf 4.451 0 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 40.867 512.277 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -512.277 cm BT /F156 9.5641 Tf 64.777 498.827 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.434 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(2)]TJ/F158 9.5641 Tf 3.858 1.434 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.434 Td [(2)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.434 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(2)]TJ/F108 8.2889 Tf 3.858 1.434 Td [(q)]TJ/F158 9.5641 Tf 4.451 0 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/:)]TJ ET 1 0 0 1 40.867 498.827 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -498.827 cm BT /F21 8.2889 Tf 52.822 476.687 Td [(Let)-250(us)-250(now)-250(write)]TJ ET 1 0 0 1 40.867 476.687 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -476.687 cm BT /F156 9.5641 Tf 64.777 459.214 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.828 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(z)]TJ/F156 6.6948 Tf 3.714 -1.435 Td [(`)]TJ/F158 9.5641 Tf 3.631 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.452 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F159 9.5641 Tf 19.165 9.085 Td [(X)]TJ/F108 5.8022 Tf -9.049 -19.542 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\025)]TJ/F21 5.8022 Tf 5.048 0 Td [(0)]TJ/F108 8.2889 Tf 4.953 10.457 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.986 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.286 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 4.355 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F108 5.8022 Tf 4.264 4.872 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F21 5.8022 Tf -3.905 -6.212 Td [(1)]TJ/F156 9.5641 Tf 9.375 2.693 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(z)]TJ/F108 5.8022 Tf 4.265 4.986 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.392 Td [(`)]TJ/F156 6.6948 Tf -3.904 -6.333 Td [(`)]TJ/F156 9.5641 Tf 7.586 2.739 Td [(;)]TJ/F21 8.2889 Tf 202.674 0 Td [(\0506.3\051)]TJ ET 1 0 0 1 40.867 459.214 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -459.214 cm BT /F21 8.2889 Tf 40.867 431.718 Td [(where)]TJ/F108 8.2889 Tf 26.783 0 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.286 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 4.355 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F21 8.2889 Tf 6.09 0 Td [(denote)-304(formal)-305(series)-304(in)-304(one)-304(formal)-305(variable)]TJ/F108 8.2889 Tf 168.943 0 Td [(q)]TJ/F21 8.2889 Tf 4.451 0 Td [(.)-305(Putting)]TJ ET 1 0 0 1 318.743 431.718 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -318.743 -431.718 cm BT /F21 8.2889 Tf 318.743 431.718 Td [(\0506.3\051)]TJ ET 1 0 0 1 336.415 431.718 cm 0 g 0 G 1 0 0 1 -336.415 -431.718 cm BT /F21 8.2889 Tf 338.937 431.718 Td [(into)]TJ ET 1 0 0 1 356.761 431.718 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -356.761 -431.718 cm BT /F21 8.2889 Tf 356.761 431.718 Td [(\0506.2\051)]TJ ET 1 0 0 1 374.433 431.718 cm 0 g 0 G 1 0 0 1 -374.433 -431.718 cm BT /F21 8.2889 Tf 376.955 431.718 Td [(gives)-304(us)-305(\050with)-304(labels)-304(analogous)-304(to)]TJ ET 1 0 0 1 40.867 431.718 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -431.718 cm BT /F21 8.2889 Tf 40.867 420.201 Td [(those)-250(we)-250(used)-250(in)-250(the)-250(previous)-250(section\051)-250(the)-250(following)-250(system:)]TJ ET 1 0 0 1 40.867 420.201 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -420.201 cm BT /F159 9.5641 Tf 69.484 409.385 Td [(X)]TJ/F108 5.8022 Tf -3.113 -19.689 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F156 6.6948 Tf 4.027 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F156 9.5641 Tf 2.497 10.603 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F108 8.2889 Tf 2.413 -3.95 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.287 Td [(I)]TJ/F156 9.5641 Tf 23.722 2.692 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F108 8.2889 Tf 3.852 -3.95 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.389 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.241 -6.688 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 19.171 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/;)]TJ/F21 8.2889 Tf 248.792 0 Td [(\0506.4\051)]TJ ET 1 0 0 1 40.867 400.299 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -400.299 cm BT /F21 8.2889 Tf 40.867 374.786 Td [(with)]TJ/F108 8.2889 Tf 19.793 0 Td [(n)]TJ/F158 9.5641 Tf 7.448 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(n)]TJ/F21 5.8022 Tf 4.791 -1.434 Td [(1)]TJ/F158 9.5641 Tf 5.982 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(n)]TJ/F156 6.6948 Tf 4.791 -1.434 Td [(`)]TJ/F21 8.2889 Tf 3.631 1.434 Td [(.)]TJ ET 1 0 0 1 40.867 374.786 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -374.786 cm BT /F21 8.2889 Tf 52.822 364.347 Td [(In)-250(the)-250(next)-250(section)-250(we)-250(prove)-250(that)-250(system)]TJ ET 1 0 0 1 209.829 364.347 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -209.829 -364.347 cm BT /F21 8.2889 Tf 209.829 364.347 Td [(\0506.4\051)]TJ ET 1 0 0 1 227.501 364.347 cm 0 g 0 G 1 0 0 1 -227.501 -364.347 cm BT /F21 8.2889 Tf 229.573 364.347 Td [(has)-250(a)-250(unique)-250(solution,)-250(thus)-250(proving)-250(the)-250(solution)-250(for)]TJ ET 1 0 0 1 426.151 364.347 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -426.151 -364.347 cm BT /F21 8.2889 Tf 426.151 364.347 Td [(\0506.2\051)]TJ ET 1 0 0 1 443.823 364.347 cm 0 g 0 G 1 0 0 1 -443.823 -364.347 cm BT /F21 8.2889 Tf 445.895 364.347 Td [(is)-250(also)-250(unique.)]TJ ET 1 0 0 1 40.867 364.347 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -364.347 cm BT /F108 8.2889 Tf 40.867 341.37 Td [(6.2.)-500(Uniqueness)-250(of)-250(the)-250(solution)]TJ ET 1 0 0 1 40.867 341.37 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -341.37 cm BT /F21 8.2889 Tf 52.822 320.554 Td [(For)-250(fixed)]TJ/F108 8.2889 Tf 35.6 0 Td [(k)]TJ/F21 8.2889 Tf 6.383 0 Td [(we)-250(have)-250(the)-250(following)-250(result:)]TJ ET 1 0 0 1 40.867 320.554 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -20.337 cm 0 g 0 G 1 0 0 1 -36.882 -300.217 cm BT /F123 8.2889 Tf 40.867 300.217 Td [(Proposition)-250(6.2.)]TJ ET 1 0 0 1 103.853 300.217 cm 0 g 0 G 1 0 0 1 -103.853 -300.217 cm BT /F108 8.2889 Tf 107.839 300.217 Td [(For)-301(all)-300(choices)-301(of)-301(nonnegative)-300(integers)-301(k)]TJ/F21 5.8022 Tf 147.154 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F108 8.2889 Tf 6.124 1.435 Td [(such)-301(that)-386(k)]TJ/F21 5.8022 Tf 41.543 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.414 1.435 Td [(C)-267(\001)-167(\001)-167(\001)-267(C)]TJ/F108 8.2889 Tf 33.754 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 7.366 1.435 Td [(D)]TJ/F108 8.2889 Tf 11.194 0 Td [(k)-301(and)-300(all)-301(nonnegative)-301(integers)]TJ ET 1 0 0 1 40.867 300.217 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -300.217 cm BT /F108 8.2889 Tf 40.867 289.756 Td [(n)]TJ/F21 5.8022 Tf 4.791 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(n)]TJ/F156 6.6948 Tf 4.791 -1.435 Td [(`)]TJ/F108 8.2889 Tf 5.703 1.435 Td [(such)-250(that)-336(n)]TJ/F108 5.8022 Tf 41.183 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.815 1.435 Td [(\025)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.435 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(,)-250(i)]TJ/F158 9.5641 Tf 8.981 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F108 8.2889 Tf 26.667 0 Td [(,)-250(the)-250(following)-250(assertion)-250(holds:)]TJ ET 1 0 0 1 40.867 289.756 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -289.756 cm BT /F108 8.2889 Tf 64.777 269.599 Td [(A)]TJ/F108 5.8022 Tf 5.031 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.286 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 9.5641 Tf 4.355 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(q)]TJ/F108 5.8022 Tf 4.452 3.95 Td [(n)]TJ/F159 9.5641 Tf 21.072 5.136 Td [(X)]TJ/F156 6.6948 Tf -15.627 -19.689 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.092 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.393 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(2)]TJ/F193 6.2485 Tf 4.211 0 Td [(B)]TJ/F108 8.2889 Tf 7.118 10.603 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.389 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.393 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.084 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.452 0 Td [(/;)]TJ/F21 8.2889 Tf 247.961 0 Td [(\0506.5\051)]TJ ET 1 0 0 1 40.867 269.599 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -269.599 cm BT /F108 8.2889 Tf 40.867 242.605 Td [(having)-250(denoted)-250(that)-335(a)]TJ/F158 9.5641 Tf 83.079 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(a)]TJ/F156 6.6948 Tf 4.418 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(,)-250(n)]TJ/F158 9.5641 Tf 11.402 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(n)]TJ/F21 5.8022 Tf 4.791 -1.435 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(n)]TJ/F156 6.6948 Tf 4.791 -1.435 Td [(`)]TJ/F108 8.2889 Tf 3.631 1.435 Td [(,)-250(and)-250(that)]TJ/F123 8.2889 Tf 37.586 0 Td [(a)]TJ/F158 9.5641 Tf 7.199 0 Td [(V)56(D)]TJ/F156 9.5641 Tf 12.215 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(a)]TJ/F156 6.6948 Tf 4.418 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 9.5641 Tf 6.225 0 Td [(2)]TJ/F193 8.9265 Tf 8.672 0 Td [(B)]TJ/F108 8.2889 Tf 9.963 0 Td [(means)-250(that)]TJ ET 1 0 0 1 442.8 242.605 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -442.8 -242.605 cm BT /F21 8.2889 Tf 442.8 242.605 Td [(\0503.2\051)]TJ ET 1 0 0 1 460.471 242.605 cm 0 g 0 G 1 0 0 1 -460.471 -242.605 cm BT /F108 8.2889 Tf 462.544 242.605 Td [(holds.)]TJ ET 1 0 0 1 40.867 242.605 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -16.184 cm 0 g 0 G 1 0 0 1 -36.882 -226.421 cm BT /F123 8.2889 Tf 40.867 226.421 Td [(Proof.)]TJ ET 1 0 0 1 65.095 226.421 cm 0 g 0 G 1 0 0 1 -65.095 -226.421 cm BT /F21 8.2889 Tf 69.08 226.421 Td [(Index)-404(the)-403(equality)]TJ ET 1 0 0 1 144.169 226.421 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -144.169 -226.421 cm BT /F21 8.2889 Tf 144.169 226.421 Td [(\0506.5\051)]TJ ET 1 0 0 1 161.841 226.421 cm 0 g 0 G 1 0 0 1 -161.841 -226.421 cm BT /F21 8.2889 Tf 165.188 226.421 Td [(by)]TJ/F156 9.5641 Tf 12.687 0 Td [(.`)]TJ/F158 9.5641 Tf 11.476 0 Td [(C)]TJ/F21 8.2889 Tf 10.892 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(-tuple)-404(of)-403(lower)-404(indices)-404(of)-404(the)-403(series)-404(appearing)-404(on)-403(the)-404(left-hand)-404(side,)-404(i.e.)-403(by)]TJ ET 1 0 0 1 40.867 226.421 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -226.421 cm BT /F156 9.5641 Tf 40.867 215.96 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.434 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(.)-250(We)-250(prove)-250(the)-250(proposition)-250(by)-250(induction)-250(on)-250(these)-250(tuples,)-250(in)-250(the)-250(usual)-250(lexicographic)-250(ordering)-250(among)-250(them.)]TJ ET 1 0 0 1 40.867 215.96 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -215.96 cm BT /F21 8.2889 Tf 52.822 205.584 Td [(Let)-212(us)-212(first)-211(check)-212(that)]TJ ET 1 0 0 1 134.733 205.584 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -134.733 -205.584 cm BT /F21 8.2889 Tf 134.733 205.584 Td [(\0506.5\051)]TJ ET 1 0 0 1 152.405 205.584 cm 0 g 0 G 1 0 0 1 -152.405 -205.584 cm BT /F21 8.2889 Tf 154.16 205.584 Td [(is)-212(true)-212(if)-211(given)-212(for)-212(smallest)-212(such)]TJ/F156 9.5641 Tf 120.717 0 Td [(.`)]TJ/F158 9.5641 Tf 9.195 0 Td [(C)]TJ/F21 8.2889 Tf 8.611 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(-tuple,)-212(that)-212(is)]TJ/F156 9.5641 Tf 52.165 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F21 8.2889 Tf 22.192 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)-212(which)-212(means)-212(that)-211(we)-212(want)]TJ ET 1 0 0 1 40.867 205.584 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -205.584 cm BT /F21 8.2889 Tf 40.867 195.123 Td [(to)-250(check)-250(if)]TJ ET 1 0 0 1 40.867 195.123 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -195.123 cm BT /F108 8.2889 Tf 64.777 178.77 Td [(A)]TJ/F108 5.8022 Tf 5.031 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F21 5.8022 Tf -20.04 -6.287 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F156 9.5641 Tf 4.832 2.693 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F159 9.5641 Tf 14.574 5.136 Td [(X)]TJ/F123 5.8022 Tf -9.129 -19.252 Td [(a)]TJ/F158 6.6948 Tf 3.18 0 Td [(2)]TJ/F193 6.2485 Tf 4.211 0 Td [(B)]TJ/F21 4.9734 Tf 5.124 -1.353 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(;:::;)]TJ/F21 4.9734 Tf 8.534 0 Td [(0)]TJ/F156 5.7385 Tf 2.879 0 Td [(;)]TJ/F108 4.9734 Tf 1.9 0 Td [(k)]TJ/F108 8.2889 Tf 4.678 11.519 Td [(A)]TJ/F108 5.8022 Tf 5.031 5.388 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.392 Td [(`)]TJ/F158 6.6948 Tf 3.184 1.392 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -50.392 -6.688 Td [(k)]TJ/F158 6.6948 Tf 3.017 0 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(a)]TJ/F156 6.6948 Tf 3.093 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 14.041 4.085 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 40.867 178.77 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -178.77 cm BT /F21 8.2889 Tf 40.867 151.914 Td [(holds.)-250(But,)-250(we)-250(see)-250(immediately)-250(that)]TJ/F193 8.9265 Tf 138.025 0 Td [(B)]TJ/F21 5.8022 Tf 7.32 -1.434 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.956 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F158 9.5641 Tf 6.172 1.434 Td [(D)-278(f)]TJ/F156 9.5641 Tf 13.655 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F21 8.2889 Tf 22.192 0 Td 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[(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(n)]TJ/F108 8.2889 Tf 3.851 -3.95 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -20.04 -6.287 Td [(k)]TJ/F156 6.6948 Tf 3.017 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 9.5641 Tf 5.175 2.693 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/;)]TJ ET 1 0 0 1 40.867 134.452 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -134.452 cm BT /F21 8.2889 Tf 40.867 118.098 Td [(which)-250(is)-250(exactly)-250(the)-250(corresponding)-250(relation)-250(in)]TJ ET 1 0 0 1 217.469 118.098 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -217.469 -118.098 cm BT /F21 8.2889 Tf 217.469 118.098 Td [(\0506.4\051)]TJ ET 1 0 0 1 235.14 118.098 cm 0 g 0 G 1 0 0 1 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1 331.631 107.722 cm 0 g 0 G 1 0 0 1 -331.631 -107.722 cm BT /F21 8.2889 Tf 333.703 107.722 Td [(corresponding)-250(to)]TJ/F108 8.2889 Tf 67.073 0 Td [(K)]TJ/F21 8.2889 Tf 8.227 0 Td [(is)]TJ ET 1 0 0 1 40.867 107.722 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -107.722 cm BT /F159 9.5641 Tf 67.89 98.331 Td [(X)]TJ/F108 5.8022 Tf -3.113 -19.69 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F156 6.6948 Tf 4.027 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F156 9.5641 Tf 2.497 10.604 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F108 8.2889 Tf 2.413 -3.95 Td [(A)]TJ/F108 5.8022 Tf 5.032 4.987 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 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Tf 18.181 25.489 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -16.735 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [([f)]TJ/F108 4.9734 Tf 5.956 0 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(g)]TJ/F21 8.2889 Tf 8.827 22.401 Td [(1)]TJ ET q 1 0 0 1 240.125 663.794 cm []0 d 0 J 0.44 w 0 0 m 10.237 0 l S Q BT /F158 9.5641 Tf 240.125 654.613 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 3.931 6.79 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.452 3.95 Td [(B)]TJ 0 -6.522 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.434 Td [(A)]TJ ET 1 0 0 1 37.134 661.403 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -661.403 cm BT /F158 9.5641 Tf 88.374 625.722 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(A)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F158 9.5641 Tf 6.145 1.434 Td [(C)]TJ/F159 9.5641 Tf 12.962 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.739 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\033)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F108 8.2889 Tf 10.773 22.39 Td [(t)]TJ/F158 9.5641 Tf 5.769 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ ET q 1 0 0 1 165.673 628.113 cm []0 d 0 J 0.44 w 0 0 m 20.153 0 l S Q BT /F158 9.5641 Tf 170.631 618.932 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 8.889 6.79 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F158 9.5641 Tf 6.145 1.434 Td [(C)]TJ/F159 9.5641 Tf 16.055 9.086 Td [(X)]TJ/F158 5.7385 Tf -5.274 -19.032 Td [(f)]TJ/F108 4.9734 Tf 2.123 0 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(g2)]TJ/F108 4.9734 Tf 5.732 0 Td [(D)]TJ/F21 4.9734 Tf 3.452 -1.705 Td [(1)]TJ/F156 5.7385 Tf 3.378 1.705 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -13.926 -6.457 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(6)1(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(A)]TJ/F159 9.5641 Tf 18.181 25.489 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -16.735 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [([f)]TJ/F108 4.9734 Tf 5.956 0 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(g)]TJ/F21 8.2889 Tf 8.827 22.401 Td [(1)]TJ ET q 1 0 0 1 283.403 628.113 cm []0 d 0 J 0.44 w 0 0 m 10.237 0 l S Q BT /F158 9.5641 Tf 283.403 618.932 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 3.931 6.79 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(:)]TJ ET 1 0 0 1 37.134 625.722 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -625.722 cm BT /F21 8.2889 Tf 37.134 596.212 Td [(The)-284(last)-283(sum)-284(above)-283(obviously)-284(goes)-283(on)-284(all)]TJ/F108 8.2889 Tf 158.478 0 Td [(B)]TJ/F158 9.5641 Tf 8.136 0 Td [(2)]TJ/F108 8.2889 Tf 9.386 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.29 0 Td [(B)]TJ/F158 9.5641 Tf 8.136 0 Td [(\033)]TJ/F108 8.2889 Tf 10.83 0 Td [(A)]TJ/F21 8.2889 Tf 5.032 0 Td [(,)-284(and)-283(every)-284(such)-283(fixed)]TJ/F108 8.2889 Tf 85.904 0 Td [(B)]TJ/F21 8.2889 Tf 7.116 0 Td [(occurs)-284(exactly)]TJ/F158 9.5641 Tf 56.755 0 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)-252(\000)-252(j)]TJ/F108 8.2889 Tf 17.752 0 Td [(A)]TJ/F158 9.5641 Tf 5.032 0 Td [(j)]TJ/F21 8.2889 Tf 5.085 0 Td [(times.)-284(Thus)-283(we)]TJ ET 1 0 0 1 37.134 596.212 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -596.212 cm BT /F21 8.2889 Tf 37.134 585.752 Td [(have)]TJ ET 1 0 0 1 37.134 585.752 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -585.752 cm BT /F156 9.5641 Tf 61.044 568.284 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.434 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 -1.434 Td [(A)]TJ/F158 9.5641 Tf 7.507 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(A)]TJ 0 -6.522 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F158 9.5641 Tf 6.145 1.434 Td [(C)]TJ/F159 9.5641 Tf 12.962 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\033)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F108 8.2889 Tf 10.773 22.389 Td [(t)]TJ/F158 9.5641 Tf 5.769 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ ET q 1 0 0 1 165.673 570.675 cm []0 d 0 J 0.44 w 0 0 m 20.153 0 l S Q BT /F158 9.5641 Tf 170.631 561.494 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 8.889 6.79 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.522 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F158 9.5641 Tf 6.145 1.434 Td [(C)]TJ/F159 9.5641 Tf 12.961 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\033)]TJ/F108 4.9734 Tf 4.477 0 Td [(A)]TJ/F158 9.5641 Tf 10.773 22.389 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)-222(\000)-222(j)]TJ/F108 8.2889 Tf 17.181 0 Td [(A)]TJ/F158 9.5641 Tf 5.031 0 Td [(j)]TJ ET q 1 0 0 1 253.374 570.675 cm []0 d 0 J 0.44 w 0 0 m 32.449 0 l S Q BT /F158 9.5641 Tf 264.481 561.494 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 15.037 6.79 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.522 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ ET 1 0 0 1 37.134 568.284 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -568.284 cm BT /F158 9.5641 Tf 88.374 538.122 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(A)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 9.5641 Tf 6.145 1.435 Td [(C)]TJ/F159 9.5641 Tf 12.962 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\033)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F108 8.2889 Tf 9.578 15.685 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 9.5641 Tf 6.677 1.435 Td [(D)]TJ/F159 9.5641 Tf 13.492 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F108 8.2889 Tf 9.579 15.685 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 9.5641 Tf 6.676 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.435 Td [(;)]TJ ET 1 0 0 1 37.134 538.122 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -538.122 cm BT /F21 8.2889 Tf 37.134 509.726 Td [(which)-250(finally)-250(proves)-250(that)]TJ/F108 8.2889 Tf 97.079 0 Td [(Ker)]TJ/F156 9.5641 Tf 13.125 0 Td [(.')]TJ/F108 5.8022 Tf 8.943 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(\022)]TJ/F108 8.2889 Tf 10.117 0 Td [(Im)]TJ/F156 9.5641 Tf 9.656 0 Td [(.')]TJ/F108 5.8022 Tf 8.943 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.012 0 Td [(t)]TJ/F158 9.5641 Tf 6.3 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(m)]TJ/F158 9.5641 Tf 9.287 0 Td [(\000)]TJ/F21 8.2889 Tf 9.586 0 Td [(2.)]TJ ET 1 0 0 1 37.134 509.726 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -509.726 cm BT /F123 8.2889 Tf 37.134 488.771 Td [(6.)-500(Recurrence)-250(relations)-250(for)-250(formal)-250(characters)]TJ ET 1 0 0 1 37.134 488.771 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -488.771 cm BT /F108 8.2889 Tf 37.134 469.549 Td [(6.1.)-500(Formal)-250(characters)-250(and)-250(systems)-250(of)-250(recurrences)]TJ ET 1 0 0 1 37.134 469.549 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -469.549 cm BT /F21 8.2889 Tf 49.089 449.572 Td [(Fix)]TJ/F108 8.2889 Tf 12.969 0 Td [(k)]TJ/F21 8.2889 Tf 5.973 0 Td [(and)]TJ/F156 9.2055 Tf 15.912 0 Td [(\003)]TJ/F158 9.5641 Tf 9.984 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F156 9.2055 Tf 3.857 1.434 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.434 Td [(0)]TJ/F158 9.5641 Tf 4.724 1.434 Td [(C)-91(\001)-166(\001)-167(\001)-90(C)]TJ/F108 8.2889 Tf 28.683 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F156 9.2055 Tf 3.631 1.434 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.434 Td [(`)]TJ/F21 8.2889 Tf 5.294 1.434 Td [(such)-201(that)]TJ/F108 8.2889 Tf 36.498 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F158 9.5641 Tf 4.724 1.434 Td [(C)-91(\001)-166(\001)-167(\001)-90(C)]TJ/F108 8.2889 Tf 28.683 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 8.2889 Tf 4.31 0 Td [(.)-201(Define)-201(the)-200(formal)-201(character)]TJ/F156 9.5641 Tf 108.787 0 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(/)]TJ/F21 8.2889 Tf 5.231 0 Td [(of)-201(a)-200(subspace)]TJ/F108 8.2889 Tf 51.497 0 Td [(W)]TJ/F158 9.5641 Tf 11.903 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ ET 1 0 0 1 37.134 449.572 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -449.572 cm BT /F21 8.2889 Tf 37.134 439.112 Td [(as)]TJ ET 1 0 0 1 37.134 439.112 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -439.112 cm BT /F156 9.5641 Tf 61.044 425.338 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(z)]TJ/F156 6.6948 Tf 3.714 -1.435 Td [(`)]TJ/F158 9.5641 Tf 3.631 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F159 9.5641 Tf 10.116 9.086 Td [(X)]TJ/F21 8.2889 Tf 15.128 -9.086 Td [(dim)]TJ/F108 8.2889 Tf 16.696 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 3.95 Td [(m)]TJ/F156 6.6948 Tf 5.013 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F108 8.2889 Tf 3.682 -2.557 Td [(q)]TJ/F108 5.8022 Tf 4.451 3.95 Td [(m)]TJ/F108 8.2889 Tf 5.511 -3.95 Td [(z)]TJ/F108 5.8022 Tf 4.265 4.873 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F21 5.8022 Tf -3.905 -6.213 Td [(1)]TJ/F158 9.5641 Tf 9.374 2.693 Td [(\001)-167(\001)-166(\001)]TJ/F108 8.2889 Tf 12.759 0 Td [(z)]TJ/F108 5.8022 Tf 4.264 4.987 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F156 6.6948 Tf -3.905 -6.333 Td [(`)]TJ/F156 9.5641 Tf 7.587 2.739 Td [(;)]TJ ET 1 0 0 1 37.134 425.338 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -425.338 cm BT /F21 8.2889 Tf 37.134 410.052 Td [(where)]TJ/F108 8.2889 Tf 26.063 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 3.472 Td [(m)]TJ/F156 6.6948 Tf 5.013 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F21 8.2889 Tf 5.483 -2.079 Td [(denotes)-217(a)-218(subspace)-217(of)]TJ/F108 8.2889 Tf 83.876 0 Td [(W)]TJ/F21 8.2889 Tf 11.048 0 Td [(spanned)-217(by)-218(monomial)-217(vectors)]TJ/F108 8.2889 Tf 115.589 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F156 6.4438 Tf 18.488 -1.434 Td [(\003)]TJ/F158 9.5641 Tf 8.284 1.434 Td [(D)]TJ/F156 9.5641 Tf 10.116 0 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.673 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\015)]TJ/F21 4.9734 Tf 3.36 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.787 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(2)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F108 8.2889 Tf 3.682 -2.079 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\015)]TJ/F156 5.7385 Tf 3.36 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.079 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.673 0 Td [(x)]TJ/F156 6.6948 Tf 4.069 -1.434 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.787 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ ET 1 0 0 1 37.134 410.052 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -410.052 cm BT /F156 9.5641 Tf 37.134 399.591 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.434 Td [(\003)]TJ/F158 9.5641 Tf 9.581 1.434 Td [(2)]TJ/F193 8.9265 Tf 9.97 0 Td [(B)]TJ/F21 8.2889 Tf 10.469 0 Td [(such)-311(that)]TJ/F108 8.2889 Tf 38.328 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 16.408 0 Td [(is)-311(of)-311(degree)]TJ/F108 8.2889 Tf 47.28 0 Td [(d)]TJ/F156 9.5641 Tf 4.609 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(//)]TJ/F158 9.5641 Tf 21.351 0 Td [(D)]TJ/F108 8.2889 Tf 11.414 0 Td [(m)]TJ/F21 8.2889 Tf 9.739 0 Td [(and)-311(weight)]TJ/F156 9.5641 Tf 45.979 0 Td [(w)-30(.)]TJ/F108 8.2889 Tf 10.884 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(//)]TJ/F158 9.5641 Tf 21.351 0 Td [(D)]TJ/F108 8.2889 Tf 11.414 0 Td [(n)]TJ/F21 5.8022 Tf 4.791 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(\015)]TJ/F21 5.8022 Tf 4.801 -1.434 Td [(1)]TJ/F158 9.5641 Tf 6.502 1.434 Td [(C)-276(\001)-167(\001)-167(\001)-276(C)]TJ/F108 8.2889 Tf 34.017 0 Td [(n)]TJ/F156 6.6948 Tf 4.791 -1.434 Td [(`)]TJ/F156 9.5641 Tf 3.632 1.434 Td [(\015)]TJ/F156 6.6948 Tf 4.801 -1.434 Td [(`)]TJ/F21 8.2889 Tf 3.631 1.434 Td [(,)-311(with)]TJ/F108 8.2889 Tf 24.818 0 Td [(d)]TJ/F156 9.5641 Tf 4.609 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(//)]TJ/F21 8.2889 Tf 19.975 0 Td [(and)]TJ/F156 9.5641 Tf 16.827 0 Td [(w)-30(.)]TJ/F108 8.2889 Tf 10.884 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(//)]TJ ET 1 0 0 1 37.134 399.591 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -399.591 cm BT /F21 8.2889 Tf 37.134 389.131 Td [(given)-250(by)]TJ ET 1 0 0 1 37.134 389.131 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -389.131 cm BT /F108 8.2889 Tf 61.044 366.407 Td [(d)]TJ/F156 9.5641 Tf 4.609 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(//)]TJ/F158 9.5641 Tf 20.054 0 Td [(D)]TJ/F158 6.6948 Tf 13.465 11.105 Td [(1)]TJ/F159 9.5641 Tf -3.349 -2.019 Td [(X)]TJ/F108 5.8022 Tf 1.638 -19.543 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(0)]TJ/F156 6.6948 Tf 11.791 21.562 Td [(`)]TJ/F159 9.5641 Tf -5.2 -2.019 Td [(X)]TJ/F108 5.8022 Tf 1.646 -19.543 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 5.006 10.457 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(j)]TJ/F158 9.5641 Tf 4.521 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F158 9.5641 Tf 5.692 0 Td [(\001)]TJ/F108 8.2889 Tf 4.784 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.435 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(\001)]TJ/F156 6.6948 Tf 1.861 0 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ ET 1 0 0 1 37.134 366.407 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -366.407 cm BT /F156 9.5641 Tf 61.044 333.009 Td [(w)-30(.)]TJ/F108 8.2889 Tf 10.884 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(//)]TJ/F158 9.5641 Tf 20.054 0 Td [(D)]TJ/F158 6.6948 Tf 13.465 11.105 Td [(1)]TJ/F159 9.5641 Tf -3.349 -2.019 Td [(X)]TJ/F108 5.8022 Tf 1.638 -19.543 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(0)]TJ/F156 6.6948 Tf 11.791 21.562 Td [(`)]TJ/F159 9.5641 Tf -5.2 -2.019 Td [(X)]TJ/F108 5.8022 Tf 1.646 -19.543 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 6.6 10.457 Td [(\015)]TJ/F108 5.8022 Tf 4.801 -1.434 Td [(i)]TJ/F158 9.5641 Tf 4.283 1.434 Td [(\001)]TJ/F108 8.2889 Tf 4.784 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.434 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(j)]TJ/F158 6.6948 Tf 1.677 0 Td [(\001)]TJ/F156 6.6948 Tf 1.861 0 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(:)]TJ ET 1 0 0 1 37.134 333.009 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -333.009 cm BT /F21 8.2889 Tf 49.089 309.716 Td [(Note)-250(that)-250(for)-250(every)-250(fixed)-250(choice)-250(of)-250(integers)]TJ/F108 8.2889 Tf 164.368 0 Td [(n)]TJ/F108 5.8022 Tf 4.791 -1.434 Td [(i)]TJ/F21 8.2889 Tf 2.157 1.434 Td [(,)]TJ/F108 8.2889 Tf 4.012 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F21 8.2889 Tf 26.668 0 Td [(,)-250(and)]TJ/F108 8.2889 Tf 20.333 0 Td [(m)]TJ/F21 8.2889 Tf 7.161 0 Td [(,)-250(such)-250(that)]TJ ET 1 0 0 1 37.134 309.716 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -309.716 cm BT /F108 8.2889 Tf 61.044 295.943 Td [(n)]TJ/F108 5.8022 Tf 4.791 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.814 1.435 Td [(\025)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F108 5.8022 Tf 4.311 -1.435 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)]TJ/F108 8.2889 Tf 13.049 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ ET 1 0 0 1 37.134 295.943 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -295.943 cm BT /F108 8.2889 Tf 61.044 282.493 Td [(m)]TJ/F158 9.5641 Tf 9.818 0 Td [(\025)]TJ/F108 8.2889 Tf 9.868 0 Td [(n)]TJ/F21 5.8022 Tf 4.791 -1.434 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(n)]TJ/F156 6.6948 Tf 4.791 -1.434 Td [(`)]TJ/F158 9.5641 Tf 5.756 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.586 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ ET 1 0 0 1 37.134 282.493 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -282.493 cm BT /F21 8.2889 Tf 37.134 268.72 Td [(and)-250(every)]TJ ET 1 0 0 1 37.134 268.72 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -268.72 cm BT /F108 8.2889 Tf 61.044 252.675 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F108 5.8022 Tf 18.487 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.183 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 6.532 2.827 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 5.388 Td [(m)]TJ/F158 6.6948 Tf 5.013 0 Td [(\000)]TJ/F156 6.6948 Tf 5.222 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(C\001\001\001C)]TJ/F108 5.8022 Tf 16.027 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.183 1.392 Td [(/)]TJ/F158 6.6948 Tf 2.498 0 Td [(\000)]TJ/F156 6.6948 Tf 5.222 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(C\001\001\001C)]TJ/F108 5.8022 Tf 16.027 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.392 Td [(/;)]TJ/F108 5.8022 Tf 4.714 0 Td [(n)]TJ/F21 4.9734 Tf 3.353 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.392 Td [(`)]TJ/F158 6.6948 Tf 3.183 1.392 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 5.8022 Tf -144.293 -6.688 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.393 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ ET 1 0 0 1 37.134 252.675 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -252.675 cm BT /F21 8.2889 Tf 37.134 237.444 Td [(we)-250(have)]TJ ET 1 0 0 1 37.134 237.444 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -237.444 cm BT /F158 9.5641 Tf 61.044 223.671 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F158 6.6948 Tf 3.185 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 9.5641 Tf 3.515 -3.95 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F108 5.8022 Tf 18.487 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.393 Td [(`)]TJ/F156 6.6948 Tf 3.183 1.393 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.828 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)]TJ/F158 6.6948 Tf 10.263 3.95 Td [(\000)]TJ/F156 9.5641 Tf 5.72 -3.95 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.828 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 5.47 -2.557 Td [(\001)-167(\001)-166(\001)]TJ/F108 8.2889 Tf 12.759 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.36 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.876 -2.597 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 9.5641 Tf 3.876 -2.597 Td [(v)]TJ/F158 9.5641 Tf 7.697 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 3.95 Td [(m)]TJ/F156 6.6948 Tf 5.013 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 -2.557 Td [(:)]TJ ET 1 0 0 1 37.134 223.671 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -223.671 cm BT /F21 8.2889 Tf 37.134 209.897 Td [(Also,)-240(if)-240(we)-239(now)-240(apply)-240(mappings)]TJ/F156 9.5641 Tf 122.598 0 Td [(')]TJ/F21 5.8022 Tf 5.376 -1.434 Td [(0)]TJ/F156 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199.436 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -199.436 cm BT /F21 8.2889 Tf 37.134 188.976 Td [(that)-250(the)-250(following)-250(equality)-250(among)-250(the)-250(dimensions)-250(of)-250(weight)-250(subspaces)-250(holds:)]TJ ET 1 0 0 1 37.134 188.976 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -188.976 cm BT /F21 8.2889 Tf 61.044 172.931 Td [(dim)]TJ/F108 8.2889 Tf 16.696 0 Td [(W)]TJ/F108 5.8022 Tf 9.247 5.388 Td [(m)]TJ/F158 6.6948 Tf 5.013 0 Td [(\000)]TJ/F156 6.6948 Tf 5.222 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(C\001\001\001C)]TJ/F108 5.8022 Tf 16.028 0 Td [(n)]TJ/F156 5.7385 Tf 3.353 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(\000)]TJ/F156 6.6948 Tf 5.222 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(C\001\001\001C)]TJ/F108 5.8022 Tf 16.027 0 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.392 Td [(`)]TJ/F158 5.7385 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[(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(dim)]TJ/F108 8.2889 Tf 16.696 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 3.95 Td [(m)]TJ/F156 6.6948 Tf 5.013 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ ET 1 0 0 1 37.134 172.931 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -172.931 cm BT /F158 9.5641 Tf 77.622 155.299 Td [(C)]TJ/F159 9.5641 Tf 15.308 9.086 Td [(X)]TJ/F108 5.8022 Tf -6.254 -19.69 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F21 4.9734 Tf 4.027 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F21 8.2889 Tf 4.091 10.604 Td [(dim)]TJ/F108 8.2889 Tf 16.696 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 4.987 Td [(m)]TJ/F156 6.6948 Tf 5.013 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -28.384 -6.286 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 9.5641 Tf 32.445 4.045 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F156 9.5641 Tf 32.46 0 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(m)]TJ/F158 6.6948 Tf 5.013 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 5.452 -3.95 Td [(dim)]TJ/F108 8.2889 Tf 16.696 0 Td [(W)]TJ/F108 5.8022 Tf 9.246 4.987 Td [(m)]TJ/F156 6.6948 Tf 5.013 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(n)]TJ/F21 4.9734 Tf 3.354 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(n)]TJ/F156 5.7385 Tf 3.354 -1.393 Td [(`)]TJ/F108 5.8022 Tf -28.385 -6.286 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(m)]TJ/F158 9.5641 Tf 32.976 3.696 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(:)]TJ ET 1 0 0 1 37.134 155.299 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -155.299 cm BT /F21 8.2889 Tf 49.089 131.799 Td [(As)-220(an)-220(implication)-220(we)-220(have)-219(the)-220(following)-220(relation)-220(among)-220(characters)-220(of)-220(Feigin\025Stoyanovsky's)-219(type)-220(subspaces)-220(at)-220(fixed)-220(level)]TJ ET 1 0 0 1 37.134 131.799 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -131.799 cm BT /F108 8.2889 Tf 37.134 121.338 Td [(k)]TJ/F21 8.2889 Tf 4.31 0 Td [(:)]TJ ET 1 0 0 1 37.134 121.338 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -121.338 cm BT /F156 9.5641 Tf 61.044 107.564 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(z)]TJ/F156 6.6948 Tf 3.714 -1.434 Td [(`)]TJ/F158 9.5641 Tf 3.631 1.434 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 5.692 0 Td [(\000)]TJ/F159 9.5641 Tf 15.84 9.086 Td [(X)]TJ/F108 5.8022 Tf -6.254 -19.689 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F21 4.9734 Tf 4.027 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F156 9.5641 Tf 4.091 10.603 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.147 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.434 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.787 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(z)]TJ/F156 6.6948 Tf 3.714 -1.434 Td [(`)]TJ/F158 9.5641 Tf 3.631 1.434 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ ET 1 0 0 1 37.134 107.564 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -107.564 cm BT /F158 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Td [(t)]TJ/F156 9.5641 Tf 3.049 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(let)-250(us)-250(define)]TJ ET 1 0 0 1 40.867 527.075 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -527.075 cm BT /F108 8.2889 Tf 64.777 502.734 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F158 9.5641 Tf 5.838 2.439 Td [(D)]TJ/F159 9.5641 Tf 16.587 9.086 Td [(X)]TJ/F158 5.7385 Tf -5.274 -19.033 Td [(f)]TJ/F108 4.9734 Tf 2.123 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g2)]TJ/F108 4.9734 Tf 5.733 0 Td [(D)]TJ/F21 4.9734 Tf 3.452 -1.705 Td [(1)]TJ/F156 5.7385 Tf 3.377 1.705 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -14.506 -6.456 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(6)1(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(I)]TJ 1.497 -1.191 Td [(t)]TJ/F159 9.5641 Tf 17.846 26.68 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -17.316 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(I)]TJ 1.497 -1.19 Td [(t)]TJ/F158 5.7385 Tf 2.683 1.19 Td [([f)]TJ/F108 4.9734 Tf 5.957 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g)]TJ/F156 9.5641 Tf 6.109 22.401 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.688 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.353 Td [(I)]TJ 1.497 -1.19 Td [(t)]TJ/F156 6.6948 Tf 3.182 2.543 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ ET q 1 0 0 1 152.562 505.125 cm []0 d 0 J 0.44 w 0 0 m 34.451 0 l S Q BT /F158 9.5641 Tf 164.669 495.943 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 16.038 6.791 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.523 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 2.573 Td [(;)]TJ ET 1 0 0 1 40.867 502.734 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -502.734 cm BT /F21 8.2889 Tf 40.867 469.337 Td [(for)]TJ/F108 8.2889 Tf 12.93 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F156 9.5641 Tf 3.182 2.439 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.316 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(j)]TJ/F158 6.6948 Tf 2.396 3.519 Td [(0)]TJ/F108 5.8022 Tf 0 -6.359 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(I)]TJ 1.497 -1.191 Td [(t)]TJ/F156 6.6948 Tf 3.182 2.543 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F21 8.2889 Tf 5.068 2.84 Td [(in)]TJ/F108 8.2889 Tf 9.582 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 9.5641 Tf 5.173 1.435 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(D)-277(f)]TJ/F108 8.2889 Tf 19.85 0 Td [(j)]TJ/F158 6.6948 Tf 2.396 3.519 Td [(0)]TJ/F21 5.8022 Tf 0 -6.212 Td [(0)]TJ/F156 9.5641 Tf 3.857 2.693 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(j)]TJ/F158 6.6948 Tf 2.396 3.471 Td [(0)]TJ/F108 5.8022 Tf 0 -6.044 Td [(t)]TJ/F158 9.5641 Tf 3.048 2.573 Td [(g)]TJ/F21 8.2889 Tf 5.611 0 Td [(for)-250(some)]TJ/F158 9.5641 Tf 35.443 0 Td [(f)]TJ/F108 8.2889 Tf 3.538 0 Td [(j)]TJ/F158 6.6948 Tf 2.396 3.519 Td [(0)]TJ/F21 5.8022 Tf 0 -6.212 Td [(0)]TJ/F156 9.5641 Tf 3.857 2.693 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(j)]TJ/F158 6.6948 Tf 2.396 3.471 Td [(0)]TJ/F108 5.8022 Tf 0 -6.044 Td [(t)]TJ/F158 9.5641 Tf 3.048 2.573 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.868 0 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(.)]TJ ET 1 0 0 1 40.867 469.337 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -469.337 cm BT /F21 8.2889 Tf 52.822 456.001 Td [(Let)-250(us)-250(prove)]TJ ET 1 0 0 1 101.212 456.001 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -101.212 -456.001 cm BT /F21 8.2889 Tf 101.212 456.001 Td [(\0505.6\051)]TJ ET 1 0 0 1 118.884 456.001 cm 0 g 0 G 1 0 0 1 -118.884 -456.001 cm BT /F21 8.2889 Tf 120.956 456.001 Td [(for)]TJ/F108 8.2889 Tf 12.931 0 Td [(A)]TJ/F158 9.5641 Tf 7.688 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(D)]TJ/F108 5.8022 Tf 5.753 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(we)-250(fixed)-250(before.)-250(We)-250(calculate:)]TJ ET 1 0 0 1 40.867 456.001 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -456.001 cm BT /F156 9.5641 Tf 64.777 437.323 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 -1.435 Td [(A)]TJ/F158 9.5641 Tf 7.506 1.435 Td [(D)]TJ/F159 9.5641 Tf 13.598 9.086 Td [(X)]TJ/F21 5.8022 Tf -2.651 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.36 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.828 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(r)]TJ/F108 8.2889 Tf 3.327 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(r)]TJ/F158 6.6948 Tf 2.922 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ ET 1 0 0 1 40.867 437.323 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -437.323 cm BT /F158 9.5641 Tf 92.106 403.996 Td [(D)]TJ/F159 9.5641 Tf 13.598 9.086 Td [(X)]TJ/F21 5.8022 Tf -2.651 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.36 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.828 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(r)]TJ/F159 9.5641 Tf 11.39 5.136 Td [(X)]TJ/F158 5.7385 Tf -5.274 -19.032 Td [(f)]TJ/F108 4.9734 Tf 2.124 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g2)]TJ/F108 4.9734 Tf 5.733 0 Td [(D)]TJ/F21 4.9734 Tf 3.451 -1.705 Td [(1)]TJ/F156 5.7385 Tf 3.378 1.705 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -19.613 -6.457 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(6)1(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(i)]TJ 1.423 -0.86 Td [(r)]TJ/F158 5.7385 Tf 2.922 0.86 Td [(g)]TJ/F159 9.5641 Tf 19.426 25.489 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -24.562 -5.865 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\023)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(i)]TJ 1.422 -0.861 Td [(r)]TJ/F158 5.7385 Tf 2.923 0.861 Td [(g)]TJ/F156 5.7385 Tf 2.123 0 Td [(/)]TJ/F158 5.7385 Tf 2.141 0 Td [([f)]TJ/F108 4.9734 Tf 5.956 0 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(g)]TJ/F156 9.5641 Tf 6.108 22.516 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.628 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.364 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(i)]TJ 1.423 -0.86 Td [(r)]TJ/F158 5.7385 Tf 2.922 0.86 Td [(g)]TJ/F156 6.6948 Tf 2.622 1.364 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ ET q 1 0 0 1 214.961 406.387 cm []0 d 0 J 0.44 w 0 0 m 44.664 0 l S Q BT /F158 9.5641 Tf 232.174 397.206 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 21.144 6.79 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.522 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.005 Td [(r)]TJ/F158 6.6948 Tf 2.922 1.005 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.498 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F21 8.2889 Tf 172.216 1.434 Td [(\0505.7\051)]TJ ET 1 0 0 1 40.867 403.996 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -403.996 cm BT /F21 8.2889 Tf 40.867 371.194 Td [(and)-285(conclude)-285(\050si)1(milarly)-285(as)-285(before\051)-285(that)-285(nontrivial)-284(summands)-285(are)-285(only)-285(those)-285(inde)1(xed)-285(by)]TJ/F108 8.2889 Tf 336.057 0 Td [(B)]TJ/F158 9.5641 Tf 8.162 0 Td [(2)]TJ/F108 8.2889 Tf 9.412 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)]TJ/F108 8.2889 Tf 4.3 0 Td [(B)]TJ/F158 9.5641 Tf 8.162 0 Td [(\023)]TJ/F156 9.5641 Tf 10.857 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(A)]TJ/F158 9.5641 Tf 7.452 0 Td [(n)-253(f)]TJ/F108 8.2889 Tf 10.599 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(r)]TJ/F158 9.5641 Tf 3.327 1.435 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(/)]TJ/F158 9.5641 Tf 5.988 0 Td [([)-253(f)]TJ/F108 8.2889 Tf 12.349 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)]TJ/F21 8.2889 Tf 3.538 0 Td [(,)-285(for)]TJ ET 1 0 0 1 40.867 371.194 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -371.194 cm BT /F21 8.2889 Tf 40.867 360.733 Td [(which)-285(also)]TJ/F108 8.2889 Tf 43.515 0 Td [(B)]TJ/F158 9.5641 Tf 8.164 0 Td [(\023)]TJ/F108 8.2889 Tf 10.859 0 Td [(A)]TJ/F21 8.2889 Tf 7.393 0 Td [(holds.)-285(We)-285(discuss)-285(two)-285(possibilities:)-285(if)]TJ/F108 8.2889 Tf 145.907 0 Td [(i)]TJ/F158 9.5641 Tf 5.769 0 Td [(D)]TJ/F108 8.2889 Tf 10.859 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(r)]TJ/F21 8.2889 Tf 3.327 1.435 Td [(,)-285(then)]TJ/F108 8.2889 Tf 24.161 0 Td [(B)]TJ/F158 9.5641 Tf 8.165 0 Td [(\023)]TJ/F108 8.2889 Tf 10.858 0 Td [(A)]TJ/F21 8.2889 Tf 7.393 0 Td [(and)-285(no)-285(additional)-285(conditions)-285(apply.)-285(But,)-285(in)-285(the)]TJ ET 1 0 0 1 40.867 360.733 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -360.733 cm BT /F21 8.2889 Tf 40.867 350.272 Td [(case)-250(of)]TJ/F108 8.2889 Tf 27.925 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(6)1(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(r)]TJ/F21 8.2889 Tf 5.399 1.435 Td [(we)-250(must)-250(request)]TJ/F108 8.2889 Tf 65.896 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.435 Td [(r)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(B)]TJ/F21 8.2889 Tf 6.838 0 Td [(in)-250(order)-250(for)]TJ/F108 8.2889 Tf 45.373 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(\023)]TJ/F108 8.2889 Tf 10.117 0 Td [(A)]TJ/F21 8.2889 Tf 7.103 0 Td [(to)-250(be)-250(true)-250(\050and)-250(then)-250(we)-250(have)]TJ/F108 8.2889 Tf 111.75 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(\023)]TJ/F108 8.2889 Tf 10.117 0 Td [(A)]TJ/F158 9.5641 Tf 7.156 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)]TJ/F21 8.2889 Tf 3.539 0 Td [(\051.)]TJ ET 1 0 0 1 40.867 350.272 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -350.272 cm BT /F21 8.2889 Tf 52.822 339.469 Td [(Therefore)]TJ ET 1 0 0 1 91.879 339.469 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -91.879 -339.469 cm BT /F21 8.2889 Tf 91.879 339.469 Td [(\0505.7\051)]TJ ET 1 0 0 1 109.551 339.469 cm 0 g 0 G 1 0 0 1 -109.551 -339.469 cm BT /F21 8.2889 Tf 111.623 339.469 Td [(can)-250(be)-250(rewritten)-250(as)-250(follows:)]TJ ET 1 0 0 1 40.867 339.469 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -339.469 cm BT /F156 9.5641 Tf 64.777 315.187 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 -1.435 Td [(A)]TJ/F158 9.5641 Tf 7.506 1.435 Td [(D)]TJ/F159 9.5641 Tf 13.598 9.086 Td [(X)]TJ/F21 5.8022 Tf -2.651 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.36 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.828 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(r)]TJ/F159 9.5641 Tf 8.297 5.136 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F156 9.5641 Tf 10.774 22.389 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.472 Td [(r)]TJ ET q 1 0 0 1 169.277 317.578 cm []0 d 0 J 0.44 w 0 0 m 22.721 0 l S Q BT /F158 9.5641 Tf 175.519 308.396 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 10.173 6.791 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 9.5641 Tf 6.145 1.435 Td [(C)]TJ/F159 9.5641 Tf 12.236 9.086 Td [(X)]TJ/F21 5.8022 Tf -2.65 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.829 0 Td [(\024)]TJ/F108 5.8022 Tf 5.047 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(r)]TJ/F159 9.5641 Tf 11.39 5.136 Td [(X)]TJ/F158 5.7385 Tf -5.274 -19.033 Td [(f)]TJ/F108 4.9734 Tf 2.123 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g2)]TJ/F108 4.9734 Tf 5.733 0 Td [(D)]TJ/F21 4.9734 Tf 3.452 -1.705 Td [(1)]TJ/F156 5.7385 Tf 3.377 1.705 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -13.925 -6.456 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(6)1(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(A)]TJ/F159 9.5641 Tf 18.181 25.489 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -16.735 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F158 5.7385 Tf 3.018 0 Td [([f)]TJ/F108 4.9734 Tf 5.957 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g)]TJ/F156 9.5641 Tf 6.109 22.401 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.629 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.364 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(i)]TJ 1.422 -0.861 Td [(r)]TJ/F158 5.7385 Tf 2.923 0.861 Td [(g)]TJ/F156 6.6948 Tf 2.621 1.364 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ ET q 1 0 0 1 332.723 317.578 cm []0 d 0 J 0.44 w 0 0 m 44.664 0 l S Q BT /F158 9.5641 Tf 349.936 308.396 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 21.145 6.791 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.523 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(r)]TJ/F158 6.6948 Tf 2.923 1.004 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ ET 1 0 0 1 40.867 315.187 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -315.187 cm BT /F158 9.5641 Tf 92.106 279.506 Td [(D)]TJ/F159 9.5641 Tf 14.323 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F159 9.5641 Tf 12.229 24.771 Td [(X)]TJ/F21 5.8022 Tf -2.65 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.829 0 Td [(\024)]TJ/F108 5.8022 Tf 5.047 0 Td [(t)]TJ/F21 8.2889 Tf 8.059 17.161 Td [(1)]TJ ET q 1 0 0 1 146.556 281.897 cm []0 d 0 J 0.44 w 0 0 m 10.237 0 l S Q BT /F158 9.5641 Tf 146.556 272.715 Td [(j)]TJ/F108 8.2889 Tf 2.735 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F108 8.2889 Tf 3.931 6.791 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.452 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.435 Td [(A)]TJ/F158 9.5641 Tf 6.146 1.435 Td [(C)]TJ/F159 9.5641 Tf 16.055 9.086 Td [(X)]TJ/F158 5.7385 Tf -5.274 -19.033 Td [(f)]TJ/F108 4.9734 Tf 2.123 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g2)]TJ/F108 4.9734 Tf 5.733 0 Td [(D)]TJ/F21 4.9734 Tf 3.451 -1.705 Td [(1)]TJ/F156 5.7385 Tf 3.378 1.705 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -13.925 -6.456 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(6)1(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(A)]TJ/F159 9.5641 Tf 18.181 25.489 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.033 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -16.735 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [([f)]TJ/F108 4.9734 Tf 5.957 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g)]TJ/F21 8.2889 Tf 8.827 22.401 Td [(1)]TJ ET q 1 0 0 1 254.369 281.897 cm []0 d 0 J 0.44 w 0 0 m 10.237 0 l S Q BT /F158 9.5641 Tf 254.369 272.715 Td [(j)]TJ/F108 8.2889 Tf 2.736 0 Td [(B)]TJ/F158 9.5641 Tf 4.766 0 Td [(j)]TJ/F159 9.5641 Tf 8.175 15.877 Td [(X)]TJ/F21 5.8022 Tf -2.65 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.829 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(r)]TJ/F158 6.6948 Tf 2.828 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.364 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(i)]TJ 1.423 -0.861 Td [(r)]TJ/F158 5.7385 Tf 2.922 0.861 Td [(g)]TJ/F156 6.6948 Tf 2.622 1.364 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F108 8.2889 Tf 2.996 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.523 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(r)]TJ/F158 6.6948 Tf 2.923 1.004 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.435 Td [(:)]TJ ET 1 0 0 1 40.867 279.506 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -279.506 cm BT /F21 8.2889 Tf 52.822 246.361 Td [(Let)-250(us)-250(now)-250(demonstrate)-250(that)-250(for)-250(every)]TJ/F158 9.5641 Tf 147.757 0 Td [(f)]TJ/F108 8.2889 Tf 3.538 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.868 0 Td [(D)]TJ/F21 5.8022 Tf 5.752 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.317 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(6)1(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(A)]TJ/F21 8.2889 Tf 7.104 0 Td [(and)-250(every)]TJ/F108 8.2889 Tf 39.546 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.317 0 Td [(B)]TJ/F158 9.5641 Tf 7.422 0 Td [(\023)]TJ/F108 8.2889 Tf 10.117 0 Td [(A)]TJ/F158 9.5641 Tf 7.157 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(g)]TJ ET 1 0 0 1 40.867 246.361 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -246.361 cm BT /F159 9.5641 Tf 69.021 234.825 Td [(X)]TJ/F21 5.8022 Tf -2.65 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(r)]TJ/F158 6.6948 Tf 2.829 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(r)]TJ/F158 6.6948 Tf 2.828 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.364 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(i)]TJ 1.423 -0.86 Td [(r)]TJ/F158 5.7385 Tf 2.922 0.86 Td [(g)]TJ/F156 6.6948 Tf 2.622 1.364 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F108 8.2889 Tf 2.996 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.523 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(r)]TJ/F158 6.6948 Tf 2.922 1.004 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 2.573 Td [(D)]TJ/F159 9.5641 Tf 16.097 9.086 Td [(X)]TJ/F21 4.9734 Tf -4.785 -18.907 Td [(0)]TJ/F158 5.7385 Tf 2.88 0 Td [(\024)]TJ/F108 4.9734 Tf 4.326 0 Td [(s)]TJ/F158 5.7385 Tf 2.03 0 Td [(\024)]TJ/F108 4.9734 Tf 4.326 0 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 4.9734 Tf -17.917 -5.635 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(6)1(D)]TJ/F108 4.9734 Tf 4.476 0 Td [(p)]TJ 2.765 -1.704 Td [(A)]TJ/F156 5.7385 Tf 3.517 1.704 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(i)]TJ/F156 5.7385 Tf 1.423 0 Td [(/)]TJ/F156 9.5641 Tf 5.642 15.456 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F158 6.6948 Tf 2.368 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.359 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(A)]TJ/F156 6.6948 Tf 3.517 1.352 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.66 0 Td [(/)]TJ/F108 8.2889 Tf 2.995 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.523 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(j)]TJ/F108 4.9734 Tf 1.677 -1.004 Td [(s)]TJ/F158 6.6948 Tf 2.527 1.004 Td [(g)]TJ/F21 8.2889 Tf 163.648 2.573 Td [(\0505.8\051)]TJ ET 1 0 0 1 40.867 225.739 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -225.739 cm BT /F21 8.2889 Tf 40.867 193.344 Td [(holds,)-250(with)]TJ/F158 9.5641 Tf 44.635 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(j)]TJ/F21 5.8022 Tf 2.395 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(j)]TJ/F108 5.8022 Tf 2.395 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.858 1.434 Td [(g)-278(D)]TJ/F108 8.2889 Tf 16.312 0 Td [(A)]TJ/F158 9.5641 Tf 7.156 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)]TJ/F21 8.2889 Tf 5.611 0 Td [(for)-250(some)]TJ/F158 9.5641 Tf 35.443 0 Td [(f)]TJ/F108 8.2889 Tf 3.538 0 Td [(j)]TJ/F21 5.8022 Tf 2.396 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(j)]TJ/F108 5.8022 Tf 2.396 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.857 1.434 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.868 0 Td [(D)]TJ/F108 5.8022 Tf 5.753 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(2)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(.)]TJ ET 1 0 0 1 40.867 193.344 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -193.344 cm BT /F21 8.2889 Tf 52.822 182.541 Td [(Since)-177(both)-176(sides)-177(of)]TJ ET 1 0 0 1 123.122 182.541 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -123.122 -182.541 cm BT /F21 8.2889 Tf 123.122 182.541 Td [(\0505.8\051)]TJ ET 1 0 0 1 140.794 182.541 cm 0 g 0 G 1 0 0 1 -140.794 -182.541 cm BT /F21 8.2889 Tf 142.258 182.541 Td [(contain)]TJ/F108 8.2889 Tf 30.052 0 Td [(t)]TJ/F158 9.5641 Tf 3.896 0 Td [(C)]TJ/F21 8.2889 Tf 7.713 0 Td [(1)-177(summands,)-176(it)-177(is)-176(enough)-177(to)-176(show)-177(the)-177(equality)-176(between)-177(the)-176(corresponding)-177(summands.)]TJ ET 1 0 0 1 40.867 182.541 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -182.541 cm BT /F21 8.2889 Tf 40.867 172.081 Td [(Depending)-250(on)-250(the)-250(choice)-250(of)-250(0)]TJ/F158 9.5641 Tf 113.561 0 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(s)]TJ/F158 9.5641 Tf 6.038 0 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(t)]TJ/F158 9.5641 Tf 5.769 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1,)]TJ/F108 8.2889 Tf 8.811 0 Td [(s)]TJ/F158 9.5641 Tf 6.039 0 Td [(6)1(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.435 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)-250(we)-250(distinguish)-250(two)-250(cases:)]TJ ET 1 0 0 1 40.867 172.081 cm 0 g 0 G 0 g 0 G 1 0 0 1 1.223 -16.469 cm 0 g 0 G 1 0 0 1 -42.09 -155.612 cm BT /F21 8.2889 Tf 42.09 155.612 Td [(1.)]TJ ET 1 0 0 1 48.837 155.612 cm 0 g 0 G 1 0 0 1 -48.837 -155.612 cm BT /F21 8.2889 Tf 52.822 155.612 Td [(For)-255(chosen)]TJ/F108 8.2889 Tf 43.312 0 Td [(s)]TJ/F156 9.5641 Tf 6.138 0 Td [(<)]TJ/F108 8.2889 Tf 10.216 0 Td [(p)]TJ/F108 5.8022 Tf 4.608 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F21 8.2889 Tf 5.679 0 Td [(the)-255(corresponding)-254(right-hand)-255(side)-255(summand)-254(equals)-255(the)-255(left-hand)-254(side)-255(summand)-255(that)-254(corresponds)]TJ ET 1 0 0 1 40.867 155.612 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -155.612 cm BT /F21 8.2889 Tf 52.822 145.151 Td [(to)]TJ/F108 8.2889 Tf 9.863 0 Td [(r)]TJ/F158 9.5641 Tf 6.698 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(s)]TJ/F21 8.2889 Tf 3.381 0 Td [(.)-250(Namely,)-250(for)]TJ/F108 8.2889 Tf 50.43 0 Td [(r)]TJ/F158 9.5641 Tf 6.697 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(s)]TJ/F156 9.5641 Tf 6.038 0 Td [(<)]TJ/F108 8.2889 Tf 10.117 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(we)-250(have)]TJ/F108 8.2889 Tf 33.794 0 Td [(p)]TJ/F108 5.8022 Tf 4.608 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(r)]TJ/F158 6.6948 Tf 2.922 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.976 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)-250(and)]TJ/F108 8.2889 Tf 23.192 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.434 Td [(r)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(j)]TJ/F108 5.8022 Tf 2.395 -1.434 Td [(s)]TJ/F21 8.2889 Tf 2.866 1.434 Td [(,)-250(and)-250(thus)]TJ ET 1 0 0 1 40.867 145.151 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -145.151 cm BT /F108 8.2889 Tf 76.732 129.367 Td [(r)]TJ/F158 9.5641 Tf 6.166 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.005 Td [(r)]TJ/F158 6.6948 Tf 2.923 1.005 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(s)]TJ/F158 9.5641 Tf 5.507 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.005 Td [(s)]TJ/F158 6.6948 Tf 2.528 1.005 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(s)]TJ/F158 9.5641 Tf 5.507 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F158 9.5641 Tf 7.456 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(s)]TJ/F158 9.5641 Tf 5.508 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F158 9.5641 Tf 6.925 0 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(p)]TJ/F108 5.8022 Tf 4.608 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.021 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ ET 1 0 0 1 40.867 129.367 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -129.367 cm BT /F156 9.5641 Tf 76.732 115.918 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(A)]TJ/F158 9.5641 Tf 7.157 0 Td [(n)-222(f)]TJ/F108 8.2889 Tf 10.303 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(r)]TJ/F158 9.5641 Tf 3.327 1.435 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(/)]TJ/F158 9.5641 Tf 5.692 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(D)]TJ/F156 9.5641 Tf 16.312 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(A)]TJ/F158 9.5641 Tf 7.157 0 Td [(n)-222(f)]TJ/F108 8.2889 Tf 10.303 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(s)]TJ/F158 9.5641 Tf 2.866 1.435 Td [(g)]TJ/F156 9.5641 Tf 3.538 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(D)]TJ/F156 9.5641 Tf 16.312 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(A)]TJ/F158 9.5641 Tf 7.157 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(n)-222(f)]TJ/F108 8.2889 Tf 10.302 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.435 Td [(s)]TJ/F158 9.5641 Tf 2.865 1.435 Td [(g)-278(D)]TJ/F156 9.5641 Tf 16.312 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(A)]TJ/F158 9.5641 Tf 7.156 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(/)]TJ/F158 9.5641 Tf 5.692 0 Td [(n)-222(f)]TJ/F108 8.2889 Tf 10.303 0 Td [(j)]TJ/F108 5.8022 Tf 2.395 -1.435 Td [(s)]TJ/F158 9.5641 Tf 2.866 1.435 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(:)]TJ ET 1 0 0 1 40.867 115.918 cm 0 g 0 G 0 g 0 G 1 0 0 1 1.223 -16.469 cm 0 g 0 G 1 0 0 1 -42.09 -99.449 cm BT /F21 8.2889 Tf 42.09 99.449 Td [(2.)]TJ ET 1 0 0 1 48.837 99.449 cm 0 g 0 G 1 0 0 1 -48.837 -99.449 cm BT /F21 8.2889 Tf 52.822 99.449 Td [(For)-251(chosen)]TJ/F108 8.2889 Tf 43.249 0 Td [(s)]TJ/F156 9.5641 Tf 6.058 0 Td [(>)]TJ/F108 8.2889 Tf 10.136 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F21 8.2889 Tf 5.647 0 Td [(the)-251(corresponding)-251(right-hand)-251(side)-251(summand)-250(equals)-251(the)-251(left-hand)-251(side)-251(summand)-251(corresponding)-251(to)]TJ ET 1 0 0 1 40.867 99.449 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -99.449 cm BT /F108 8.2889 Tf 52.822 88.988 Td [(r)]TJ/F158 9.5641 Tf 6.697 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(s)]TJ/F158 9.5641 Tf 5.507 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1,)-250(because)]TJ/F108 8.2889 Tf 41.345 0 Td [(s)]TJ/F158 9.5641 Tf 6.039 0 Td [(\025)]TJ/F108 8.2889 Tf 9.868 0 Td [(p)]TJ/F108 5.8022 Tf 4.608 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F158 9.5641 Tf 7.456 0 Td [(\051)]TJ/F108 8.2889 Tf 12.632 0 Td [(r)]TJ/F158 9.5641 Tf 6.698 0 Td [(\025)]TJ/F108 8.2889 Tf 9.868 0 Td [(p)]TJ/F108 5.8022 Tf 4.608 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(implies)-250(that)]TJ/F108 8.2889 Tf 47.793 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(r)]TJ/F158 6.6948 Tf 2.923 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(p)]TJ/F108 5.8022 Tf 4.608 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.021 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(and)]TJ/F108 8.2889 Tf 16.321 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.434 Td [(r)]TJ/F158 9.5641 Tf 5.984 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.434 Td [(s)]TJ/F158 6.6948 Tf 2.367 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.515 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(j)]TJ/F108 5.8022 Tf 2.396 -1.434 Td [(s)]TJ/F21 8.2889 Tf 2.865 1.434 Td [(.)-250(Therefore:)]TJ ET 1 0 0 1 40.867 88.988 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -88.988 cm BT /F108 8.2889 Tf 76.732 73.204 Td 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G 1 0 0 1 -37.134 -702.503 cm BT /F108 6.6311 Tf 169.047 702.503 Td [(M.)-250(Jerkovi\242)-250(/)-250(Journal)-250(of)-250(Pure)-250(and)-250(Applied)-250(Algebra)-250(213)-250(\0502009\051)-250(913\025926)]TJ/F21 6.6311 Tf 322.82 0 Td [(921)]TJ ET 1 0 0 1 503.385 702.503 cm 0 g 0 G 1 0 0 1 -503.385 -702.503 cm BT /F21 8.2889 Tf 49.089 680.585 Td [(We)-268(continue)-267(the)-268(proof)-268(by)-267(showing)-268(that)]TJ/F108 8.2889 Tf 152.539 0 Td [(Im)]TJ/F156 9.5641 Tf 9.657 0 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.434 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.434 Td [(/)]TJ/F158 9.5641 Tf 6.6 0 Td [(D)]TJ/F108 8.2889 Tf 10.493 0 Td [(Ker)]TJ/F156 9.5641 Tf 13.126 0 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.159 0 Td [(t)]TJ/F158 9.5641 Tf 6.676 0 Td [(D)]TJ/F21 8.2889 Tf 10.493 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(m)]TJ/F158 9.5641 Tf 9.437 0 Td [(\000)]TJ/F21 8.2889 Tf 9.736 0 Td [(2.)-268(First)-267(we)-268(show)-268(that)]TJ/F108 8.2889 Tf 82.255 0 Td [(Im)]TJ/F156 9.5641 Tf 9.657 0 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.434 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.434 Td [(/)]TJ/F158 9.5641 Tf 6.601 0 Td [(\022)]TJ/F108 8.2889 Tf 10.493 0 Td [(Ker)]TJ/F156 9.5641 Tf 13.125 0 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ ET 1 0 0 1 37.134 680.585 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -680.585 cm BT /F21 8.2889 Tf 37.134 670.125 Td [(by)-250(proving)-250(that)]TJ ET 1 0 0 1 37.134 670.125 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -670.125 cm BT /F156 9.5641 Tf 61.044 653.768 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(.w)-30(//)]TJ/F158 9.5641 Tf 20.676 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ ET 1 0 0 1 37.134 653.768 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -653.768 cm BT /F21 8.2889 Tf 37.134 637.411 Td [(for)-250(every)]TJ ET 1 0 0 1 37.134 637.411 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -637.411 cm BT /F156 9.5641 Tf 61.044 619.684 Td [(w)]TJ/F158 9.5641 Tf 9.973 0 Td [(D)]TJ/F159 9.5641 Tf 15.678 9.086 Td [(X)]TJ/F108 5.8022 Tf -5.561 -19.69 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.683 1.004 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.004 Td [(t)]TJ/F156 6.6948 Tf 2.684 1.004 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F156 9.5641 Tf 4.091 10.604 Td [(w)]TJ/F108 5.8022 Tf 7.03 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F158 9.5641 Tf 5.839 2.439 Td [(2)]TJ/F159 9.5641 Tf 14.233 9.086 Td [(X)]TJ/F108 5.8022 Tf -5.56 -19.69 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.004 Td [(t)]TJ/F156 6.6948 Tf 2.684 1.004 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.091 10.604 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F156 9.5641 Tf 3.182 2.439 Td [(;)]TJ ET 1 0 0 1 37.134 619.684 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -619.684 cm BT /F108 8.2889 Tf 37.134 593.39 Td [(t)]TJ/F158 9.5641 Tf 6.299 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(m)]TJ/F158 9.5641 Tf 9.287 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(2.)-250(For)-250(all)]TJ/F108 8.2889 Tf 34.532 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(2)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(D)-278(f)]TJ/F108 8.2889 Tf 13.655 0 Td [(i)]TJ/F21 5.8022 Tf 2.371 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.857 1.435 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.868 0 Td [(D)]TJ/F108 5.8022 Tf 5.753 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(2)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(the)-250(following)-250(holds:)]TJ ET 1 0 0 1 37.134 593.39 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -593.39 cm BT /F156 9.5641 Tf 61.044 575.421 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.434 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.434 Td [(.w)-30(//)]TJ/F108 5.8022 Tf 18.019 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 9.5641 Tf 7.363 2.787 Td [(D)]TJ/F159 9.5641 Tf 19.346 9.086 Td [(X)]TJ/F21 5.8022 Tf -8.399 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.36 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.876 -2.597 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(.w)-30(//)]TJ/F108 5.8022 Tf 18.019 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.03 -1.705 Td [(1)]TJ/F158 6.6948 Tf 3.875 2.709 Td [(g)]TJ/F156 9.5641 Tf 2.976 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ ET 1 0 0 1 37.134 575.421 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -575.421 cm BT /F158 9.5641 Tf 123.829 549.478 Td [(D)]TJ/F159 9.5641 Tf 24.83 9.086 Td [(X)]TJ/F21 5.8022 Tf -13.883 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(2)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(<)]TJ/F108 5.8022 Tf 5.222 0 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.36 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 -2.597 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(2)]TJ/F108 8.2889 Tf 3.876 -2.597 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(1)]TJ/F156 6.6948 Tf 3.876 2.709 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(2)]TJ/F158 6.6948 Tf 3.876 2.709 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 9.5641 Tf 6.001 2.788 Td [(C)]TJ/F159 9.5641 Tf 23.468 9.086 Td [(X)]TJ/F21 5.8022 Tf -13.882 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(<)]TJ/F108 5.8022 Tf 5.222 0 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.359 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.875 -2.597 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(2)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F108 8.2889 Tf 3.858 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(1)]TJ/F156 6.6948 Tf 3.876 2.709 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(2)]TJ/F158 6.6948 Tf 3.876 2.709 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ ET 1 0 0 1 37.134 549.478 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -549.478 cm BT /F158 9.5641 Tf 123.829 523.534 Td [(D)]TJ/F159 9.5641 Tf 24.83 9.086 Td [(X)]TJ/F21 5.8022 Tf -13.883 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(2)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(<)]TJ/F108 5.8022 Tf 5.222 0 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.36 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 -2.597 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(2)]TJ/F108 8.2889 Tf 3.876 -2.597 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(1)]TJ/F156 6.6948 Tf 3.876 2.709 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(2)]TJ/F158 6.6948 Tf 3.876 2.709 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 9.5641 Tf 6.001 2.788 Td [(\000)]TJ/F159 9.5641 Tf 23.468 9.086 Td [(X)]TJ/F21 5.8022 Tf -13.882 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(<)]TJ/F108 5.8022 Tf 5.222 0 Td [(s)]TJ/F21 4.9734 Tf 2.367 -1.353 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.359 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.875 -2.597 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F21 4.9734 Tf 2.368 -1.353 Td [(2)]TJ/F108 8.2889 Tf 3.876 -2.597 Td [(a)]TJ/F156 9.5641 Tf 4.417 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.452 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(1)]TJ/F156 6.6948 Tf 3.876 2.709 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(s)]TJ/F21 4.9734 Tf 2.029 -1.705 Td [(2)]TJ/F158 6.6948 Tf 3.875 2.709 Td [(g)]TJ/F156 9.5641 Tf 2.976 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(2)]TJ/F158 9.5641 Tf 6.532 2.788 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ ET 1 0 0 1 37.134 523.534 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -523.534 cm BT /F21 8.2889 Tf 37.134 497.199 Td [(hence)]TJ/F156 9.5641 Tf 24.808 0 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.049 1.435 Td [(.w)-30(//)]TJ/F158 9.5641 Tf 20.675 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)-250(follows.)]TJ ET 1 0 0 1 37.134 497.199 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -497.199 cm BT /F21 8.2889 Tf 49.089 486.632 Td [(Next,)-250(we)-250(prove)-250(that)]TJ/F108 8.2889 Tf 77.277 0 Td [(Ker)]TJ/F156 9.5641 Tf 13.125 0 Td [(.')]TJ/F108 5.8022 Tf 8.943 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(\022)]TJ/F108 8.2889 Tf 10.117 0 Td [(Im)]TJ/F156 9.5641 Tf 9.657 0 Td [(.')]TJ/F108 5.8022 Tf 8.942 -1.435 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(.)-250(Take)]TJ ET 1 0 0 1 37.134 486.632 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -486.632 cm BT /F156 9.5641 Tf 61.044 468.663 Td [(w)]TJ/F158 9.5641 Tf 9.973 0 Td [(D)]TJ/F159 9.5641 Tf 23.033 9.086 Td [(X)]TJ/F108 5.8022 Tf -12.916 -19.689 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.477 0 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F156 9.5641 Tf 4.092 10.603 Td [(w)]TJ/F108 5.8022 Tf 7.029 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 6.533 2.787 Td [(2)]TJ/F159 9.5641 Tf 21.589 9.086 Td [(X)]TJ/F108 5.8022 Tf -12.917 -19.689 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.091 10.603 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ ET 1 0 0 1 37.134 468.663 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -468.663 cm BT /F21 8.2889 Tf 37.134 441.705 Td [(such)-250(that)]TJ/F156 9.5641 Tf 37.316 0 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.w)-30(/)]TJ/F158 9.5641 Tf 17.108 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0.)-250(This)-250(means)-250(that)-250(for)-250(every)]TJ/F108 8.2889 Tf 107.58 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.515 1.435 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.551 0 Td [(C)]TJ/F21 5.8022 Tf 5.221 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(and)]TJ/F158 9.5641 Tf 16.32 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.867 0 Td [(D)]TJ/F21 5.8022 Tf 5.753 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.316 0 Td [(i)]TJ/F158 9.5641 Tf 5.028 0 Td [(6)1(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 5.93 1.435 Td [(one)-250(has)]TJ ET 1 0 0 1 37.134 441.705 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -441.705 cm BT /F156 9.5641 Tf 61.044 425.348 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.435 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(:)]TJ ET 1 0 0 1 37.134 425.348 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -425.348 cm BT /F21 8.2889 Tf 37.134 408.991 Td [(Fix)]TJ/F108 8.2889 Tf 13.598 0 Td [(A)]TJ/F158 9.5641 Tf 8.251 0 Td [(D)-337(f)]TJ/F108 8.2889 Tf 14.218 0 Td [(i)]TJ/F21 5.8022 Tf 2.371 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.435 Td [(t)]TJ/F158 9.5641 Tf 3.048 1.435 Td [(g)-337(2)]TJ/F108 8.2889 Tf 15.994 0 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 5.86 0 Td [(and)-276(arbitrary)]TJ/F158 9.5641 Tf 52.534 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(g)-337(2)]TJ/F108 8.2889 Tf 15.994 0 Td [(D)]TJ/F21 5.8022 Tf 5.753 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)]TJ/F108 8.2889 Tf 4.231 0 Td [(i)]TJ/F158 9.5641 Tf 5.59 0 Td [(6)1(2)]TJ/F108 8.2889 Tf 9.236 0 Td [(A)]TJ/F21 8.2889 Tf 5.031 0 Td [(.)-277(We)-276(have)]TJ/F108 8.2889 Tf 39.991 0 Td [(A)]TJ/F158 9.5641 Tf 7.382 0 Td [([)-246(f)]TJ/F108 8.2889 Tf 12.278 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(g)-337(D)-336(f)]TJ/F108 8.2889 Tf 20.977 0 Td [(j)]TJ/F21 5.8022 Tf 2.396 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(j)]TJ/F108 5.8022 Tf 2.395 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.858 1.435 Td [(g)]TJ/F21 8.2889 Tf 5.83 0 Td [(for)-277(some)]TJ/F158 9.5641 Tf 35.883 0 Td [(f)]TJ/F108 8.2889 Tf 3.538 0 Td [(j)]TJ/F21 5.8022 Tf 2.396 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(j)]TJ/F108 5.8022 Tf 2.396 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.857 1.435 Td [(g)-337(2)]TJ ET 1 0 0 1 37.134 408.991 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -408.991 cm BT /F108 8.2889 Tf 37.134 398.53 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(2)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)-250(so)]TJ/F108 8.2889 Tf 14.481 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(j)]TJ/F108 5.8022 Tf 2.395 -1.435 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(A)]TJ/F156 6.6948 Tf 3.517 1.352 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.66 0 Td [(/)]TJ/F21 8.2889 Tf 5.067 1.435 Td [(for)-250(some)]TJ/F108 8.2889 Tf 35.443 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.435 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(i)]TJ/F156 9.5641 Tf 2.371 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(2)-278(f)]TJ/F21 8.2889 Tf 12.211 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(t)]TJ/F158 9.5641 Tf 5.768 0 Td [(C)]TJ/F21 8.2889 Tf 9.586 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)]TJ/F21 8.2889 Tf 3.539 0 Td [(.)]TJ ET 1 0 0 1 37.134 398.53 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -398.53 cm BT /F21 8.2889 Tf 49.089 387.964 Td [(From)]TJ/F156 9.5641 Tf 22.073 0 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.435 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)-250(and)]TJ ET 1 0 0 1 37.134 387.964 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -387.964 cm BT /F156 9.5641 Tf 61.044 369.279 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F158 9.5641 Tf 6.462 1.435 Td [(D)]TJ/F159 9.5641 Tf 17.657 9.085 Td [(X)]TJ/F21 5.8022 Tf -6.71 -19.542 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F158 6.6948 Tf 2.367 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.36 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F108 8.2889 Tf 2.865 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F156 6.6948 Tf 14.451 -1.435 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(j)]TJ/F108 4.9734 Tf 1.677 -1.004 Td [(s)]TJ/F158 6.6948 Tf 2.527 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.976 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ ET 1 0 0 1 37.134 369.279 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -369.279 cm BT /F158 9.5641 Tf 111.093 343.791 Td [(D)]TJ/F156 9.5641 Tf 10.947 0 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(A)]TJ/F156 6.6948 Tf 3.517 1.352 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F108 8.2889 Tf 2.996 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(A)]TJ/F156 9.5641 Tf 4.02 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F158 9.5641 Tf 5.101 1.435 Td [(C)]TJ/F159 9.5641 Tf 15.566 9.086 Td [(X)]TJ/F21 4.9734 Tf -4.785 -18.907 Td [(0)]TJ/F158 5.7385 Tf 2.879 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 4.9734 Tf -17.918 -5.635 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(6)1(D)]TJ/F108 4.9734 Tf 4.476 0 Td [(p)]TJ 2.766 -1.704 Td [(A)]TJ/F156 5.7385 Tf 3.516 1.704 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(i)]TJ/F156 5.7385 Tf 1.422 0 Td [(/)]TJ/F156 9.5641 Tf 5.643 15.456 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F108 8.2889 Tf 2.865 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F156 6.6948 Tf 14.452 -1.435 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.498 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(j)]TJ/F108 4.9734 Tf 1.676 -1.004 Td [(s)]TJ/F158 6.6948 Tf 2.528 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.435 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ ET 1 0 0 1 37.134 343.791 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -343.791 cm BT /F21 8.2889 Tf 37.134 312.314 Td [(the)-250(following)-250(equation)-250(holds:)]TJ ET 1 0 0 1 37.134 312.314 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -312.314 cm BT /F108 8.2889 Tf 61.044 292.595 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.434 Td [(A)]TJ/F156 9.5641 Tf 4.021 1.434 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.434 Td [(D)]TJ/F159 9.5641 Tf 16.097 9.086 Td [(X)]TJ/F21 4.9734 Tf -4.785 -18.907 Td [(0)]TJ/F158 5.7385 Tf 2.879 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 4.9734 Tf -17.918 -5.635 Td [(s)]TJ/F158 5.7385 Tf 2.03 0 Td [(6)1(D)]TJ/F108 4.9734 Tf 4.476 0 Td [(p)]TJ 2.765 -1.704 Td [(A)]TJ/F156 5.7385 Tf 3.517 1.704 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(i)]TJ/F156 5.7385 Tf 1.423 0 Td [(/)]TJ/F156 9.5641 Tf 5.642 15.456 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F158 6.6948 Tf 2.367 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.36 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(A)]TJ/F156 6.6948 Tf 3.517 1.352 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F108 8.2889 Tf 2.996 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F156 6.6948 Tf 14.451 -1.434 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(j)]TJ/F108 4.9734 Tf 1.677 -1.005 Td [(s)]TJ/F158 6.6948 Tf 2.527 1.005 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.434 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.434 Td [(:)]TJ/F21 8.2889 Tf 223.151 0 Td [(\0505.3\051)]TJ ET 1 0 0 1 37.134 292.595 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -292.595 cm BT /F21 8.2889 Tf 49.089 261.013 Td [(By)]TJ ET 1 0 0 1 60.71 261.013 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -60.71 -261.013 cm BT /F21 8.2889 Tf 60.71 261.013 Td [(Lemma)-250(5.6)]TJ ET 1 0 0 1 102.486 261.013 cm 0 g 0 G 1 0 0 1 -102.486 -261.013 cm BT /F21 8.2889 Tf 104.558 261.013 Td [(we)-250(can)-250(write)]TJ ET 1 0 0 1 37.134 261.013 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -261.013 cm BT /F108 8.2889 Tf 61.044 243.286 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.434 Td [(A)]TJ/F158 9.5641 Tf 6.677 1.434 Td [(D)]TJ/F159 9.5641 Tf 13.493 9.086 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.738 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F108 8.2889 Tf 9.578 15.684 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.522 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.572 Td [(;)]TJ ET 1 0 0 1 37.134 243.286 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -243.286 cm BT /F21 8.2889 Tf 37.134 210.906 Td [(where)]TJ/F108 8.2889 Tf 26.231 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.52 Td [(B)]TJ 0 -6.212 Td [(A)]TJ/F21 8.2889 Tf 5.989 2.692 Td [(denotes)-238(the)-237(part)-238(of)]TJ/F108 8.2889 Tf 73.715 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.452 -1.435 Td [(A)]TJ/F21 8.2889 Tf 5.989 1.435 Td [(in)]TJ/F193 8.9265 Tf 9.479 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F21 8.2889 Tf 3.834 -3.472 Td [(.)-238(Analogous)-237(decomposition)-238(holds)-237(also)-238(for)-237(summands)-238(on)-238(the)-237(right-hand)-238(side)-237(of)]TJ ET 1 0 0 1 37.134 210.906 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -10.461 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -37.134 -200.445 cm BT /F21 8.2889 Tf 37.134 200.445 Td [(\0505.3\051)]TJ ET 1 0 0 1 54.806 200.445 cm 0 g 0 G 1 0 0 1 -54.806 -200.445 cm BT /F21 8.2889 Tf 54.806 200.445 Td [(.)-250(Introducing)-250(these)-250(expressions)-250(into)]TJ ET 1 0 0 1 192.591 200.445 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -192.591 -200.445 cm BT /F21 8.2889 Tf 192.591 200.445 Td [(\0505.3\051)]TJ ET 1 0 0 1 210.263 200.445 cm 0 g 0 G 1 0 0 1 -210.263 -200.445 cm BT /F21 8.2889 Tf 212.335 200.445 Td [(gives)]TJ ET 1 0 0 1 37.134 200.445 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -200.445 cm BT /F159 9.5641 Tf 66.014 189.812 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.739 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F108 8.2889 Tf 9.579 15.685 Td [(a)]TJ/F156 9.5641 Tf 4.417 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.452 3.95 Td [(B)]TJ 0 -6.523 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.521 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.435 Td [(D)]TJ/F159 9.5641 Tf 16.097 9.086 Td [(X)]TJ/F21 4.9734 Tf -4.785 -18.907 Td [(0)]TJ/F158 5.7385 Tf 2.88 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 4.9734 Tf -17.917 -5.635 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(6)1(D)]TJ/F108 4.9734 Tf 4.476 0 Td [(p)]TJ 2.765 -1.704 Td [(A)]TJ/F156 5.7385 Tf 3.517 1.704 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(i)]TJ/F156 5.7385 Tf 1.423 0 Td [(/)]TJ/F156 9.5641 Tf 5.642 15.456 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F158 6.6948 Tf 2.368 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.359 0 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(A)]TJ/F156 6.6948 Tf 3.517 1.352 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.66 0 Td [(/)]TJ/F159 9.5641 Tf 16.584 5.136 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -24.373 -5.865 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F156 5.7385 Tf 4.476 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(A)]TJ/F158 5.7385 Tf 3.018 0 Td [([f)]TJ/F108 4.9734 Tf 5.957 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g)]TJ/F156 5.7385 Tf 2.124 0 Td [(/)]TJ/F158 5.7385 Tf 2.14 0 Td [(nf)]TJ/F108 4.9734 Tf 4.906 0 Td [(j)]TJ 1.438 -0.861 Td [(s)]TJ/F158 5.7385 Tf 2.527 0.861 Td [(g)]TJ/F108 8.2889 Tf 4.913 15.811 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.523 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(j)]TJ/F108 4.9734 Tf 1.677 -1.004 Td [(s)]TJ/F158 6.6948 Tf 2.527 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.975 2.573 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(:)]TJ/F21 8.2889 Tf 158.967 0 Td [(\0505.4\051)]TJ ET 1 0 0 1 37.134 180.726 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -180.726 cm BT /F21 8.2889 Tf 49.089 147.581 Td [(Some)-354(of)-354(the)-354(summands)-354(in)]TJ ET 1 0 0 1 154.299 147.581 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -154.299 -147.581 cm BT /F21 8.2889 Tf 154.299 147.581 Td [(\0505.4\051)]TJ ET 1 0 0 1 171.971 147.581 cm 0 g 0 G 1 0 0 1 -171.971 -147.581 cm BT /F21 8.2889 Tf 174.905 147.581 Td [(are)-354(trivial.)-355(We)-354(want)-354(to)-354(s)1(ee)-354(for)-354(which)]TJ/F108 8.2889 Tf 147.421 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 18.697 0 Td [(2)]TJ/F193 8.9265 Tf 10.882 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F21 8.2889 Tf 6.769 -3.472 Td [(\050)]TJ/F108 8.2889 Tf 3.067 0 Td [(B)]TJ/F158 9.5641 Tf 9.632 0 Td [(2)]TJ/F108 8.2889 Tf 10.883 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 6.502 0 Td [(such)-354(that)]TJ/F108 8.2889 Tf 39.04 0 Td [(B)]TJ/F158 9.5641 Tf 9.633 0 Td [(\023)]TJ/F108 8.2889 Tf 12.327 0 Td [(A)]TJ/F21 8.2889 Tf 5.031 0 Td [(\051)-354(also)]TJ ET 1 0 0 1 37.134 147.581 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -147.581 cm BT /F108 8.2889 Tf 37.134 137.121 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F21 8.2889 Tf 5.047 1.435 Td [(holds.)-250(There)-250(are)-250(two)-250(cases:)]TJ ET 1 0 0 1 37.134 137.121 cm 0 g 0 G 0 g 0 G 1 0 0 1 1.223 -15.76 cm 0 g 0 G 1 0 0 1 -38.357 -121.361 cm BT /F21 8.2889 Tf 38.357 121.361 Td [(1.)]TJ ET 1 0 0 1 45.104 121.361 cm 0 g 0 G 1 0 0 1 -45.104 -121.361 cm BT /F108 8.2889 Tf 49.089 121.361 Td [(A)]TJ/F158 9.5641 Tf 7.157 0 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(g)-278(\022)]TJ/F108 8.2889 Tf 16.312 0 Td [(B)]TJ/F21 8.2889 Tf 4.766 0 Td [(:)-250(because)-250(of)]TJ ET 1 0 0 1 138.803 121.361 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -138.803 -121.361 cm BT /F21 8.2889 Tf 138.803 121.361 Td [(Lemma)-250(5.3)]TJ ET 1 0 0 1 180.579 121.361 cm 0 g 0 G 1 0 0 1 -180.579 -121.361 cm BT /F21 8.2889 Tf 182.651 121.361 Td [(the)-250(statement)-250(obviously)-250(holds.)]TJ ET 1 0 0 1 37.134 121.361 cm 0 g 0 G 0 g 0 G 1 0 0 1 1.223 -10.672 cm 0 g 0 G 1 0 0 1 -38.357 -110.689 cm BT /F21 8.2889 Tf 38.357 110.689 Td [(2.)]TJ ET 1 0 0 1 45.104 110.689 cm 0 g 0 G 1 0 0 1 -45.104 -110.689 cm BT /F108 8.2889 Tf 49.089 110.689 Td [(A)]TJ/F158 9.5641 Tf 7.122 0 Td [([)-219(f)]TJ/F108 8.2889 Tf 12.018 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(6)1(\022)]TJ/F108 8.2889 Tf 16.312 0 Td [(B)]TJ/F21 8.2889 Tf 4.766 0 Td [(:)-249(let)]TJ/F108 8.2889 Tf 16.969 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F21 8.2889 Tf 5.895 -3.472 Td [(and)-249(let)-248(us)-249(suppose)]TJ/F108 8.2889 Tf 72.449 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.521 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F21 8.2889 Tf 2.975 1.434 Td [(.)-249(Then)]TJ/F108 8.2889 Tf 25.101 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F158 9.5641 Tf 5.924 1.434 Td [(\134)]TJ/F193 8.9265 Tf 8.48 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.521 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.434 Td [(D)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g[)]TJ/F108 5.8022 Tf 6.949 0 Td [(B)]TJ/F21 8.2889 Tf 3.834 1.434 Td [(.)-249(Since)]TJ/F108 8.2889 Tf 26.27 0 Td [(A)]TJ/F158 9.5641 Tf 7.122 0 Td [([)-219(f)]TJ/F108 8.2889 Tf 12.018 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-219([)]TJ/F108 8.2889 Tf 14.108 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(\033)]TJ/F108 8.2889 Tf 10.117 0 Td [(B)]TJ/F21 8.2889 Tf 4.766 0 Td [(,)]TJ ET 1 0 0 1 37.134 110.689 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -110.689 cm BT /F21 8.2889 Tf 49.089 99.063 Td [(this)-250(is)-250(in)-250(opposition)-250(to)]TJ/F108 8.2889 Tf 86.942 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.672 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.471 Td [(B)]TJ/F21 8.2889 Tf 3.835 -3.471 Td [(,)-250(and)-250(thus)]TJ/F108 8.2889 Tf 39.198 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.487 0 Td [(6)1(2)]TJ/F193 8.9265 Tf 8.672 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F21 8.2889 Tf 2.975 1.435 Td [(.)]TJ ET 1 0 0 1 37.134 99.063 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -99.063 cm BT /F21 8.2889 Tf 49.089 83.409 Td [(Now)]TJ ET 1 0 0 1 69.09 83.409 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -69.09 -83.409 cm BT /F21 8.2889 Tf 69.09 83.409 Td [(\0505.4\051)]TJ ET 1 0 0 1 86.762 83.409 cm 0 g 0 G 1 0 0 1 -86.762 -83.409 cm BT /F21 8.2889 Tf 88.834 83.409 Td [(becomes)]TJ ET 1 0 0 1 37.134 83.409 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -83.409 cm BT /F159 9.5641 Tf 65.401 74.768 Td [(X)]TJ/F108 4.9734 Tf -2.18 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -16.735 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.859 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F158 5.7385 Tf 3.019 0 Td [([f)]TJ/F108 4.9734 Tf 5.957 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g)]TJ/F108 8.2889 Tf 4.913 15.696 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ 0 -6.522 Td [(A)]TJ/F156 9.5641 Tf 4.02 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F158 9.5641 Tf 5.631 1.434 Td [(D)]TJ/F159 9.5641 Tf 16.098 9.086 Td [(X)]TJ/F21 4.9734 Tf -4.785 -18.906 Td [(0)]TJ/F158 5.7385 Tf 2.879 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(\024)]TJ/F108 4.9734 Tf 4.327 0 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F108 4.9734 Tf -17.918 -5.636 Td [(s)]TJ/F158 5.7385 Tf 2.029 0 Td [(6)1(D)]TJ/F108 4.9734 Tf 4.476 0 Td [(p)]TJ 2.765 -1.704 Td [(A)]TJ/F156 5.7385 Tf 3.517 1.704 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(i)]TJ/F156 5.7385 Tf 1.422 0 Td [(/)]TJ/F156 9.5641 Tf 5.643 15.456 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(s)]TJ/F158 6.6948 Tf 2.367 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.36 0 Td [(C)]TJ/F108 5.8022 Tf 5.221 0 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(A)]TJ/F156 6.6948 Tf 3.517 1.352 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F159 9.5641 Tf 8.947 5.136 Td [(X)]TJ/F108 4.9734 Tf -2.181 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(D)]TJ/F156 5.7385 Tf 3.452 0 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -16.735 -5.75 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F158 5.7385 Tf 3.018 0 Td [([f)]TJ/F108 4.9734 Tf 5.957 0 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(g)]TJ/F108 8.2889 Tf 4.913 15.696 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 3.95 Td [(B)]TJ/F156 6.6948 Tf 0 -6.522 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(A)]TJ/F158 6.6948 Tf 3.521 0 Td [([f)]TJ/F108 5.8022 Tf 6.95 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 6.6948 Tf 2.477 0 Td [(/)]TJ/F158 6.6948 Tf 2.497 0 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(j)]TJ/F108 4.9734 Tf 1.677 -1.005 Td [(s)]TJ/F158 6.6948 Tf 2.527 1.005 Td [(g)]TJ/F156 9.5641 Tf 2.976 2.572 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.434 Td [(A)]TJ/F158 6.6948 Tf 3.522 0 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.434 Td [(;)]TJ ET 1 0 0 1 37.134 65.682 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -25.196 cm 0 g 0 G 1 0 0 1 466.251 0 cm 0 g 0 G endstream endobj 163 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./main.pdf) /PTEX.PageNumber 8 /PTEX.InfoDict 124 0 R /Matrix [ 1 0 0 1 0 0 ] /BBox [ 0 0 544.252 742.677 ] /Resources << /Font << /F123 121 0 R /F156 120 0 R /F108 115 0 R /F21 117 0 R /F158 113 0 R /F193 43 0 R /F159 112 0 R /F137 39 0 R >> /ProcSet [ /PDF /Text ] >> /Length 40178 >> stream 1 0 0 1 40.867 702.503 cm 0 g 0 G 1 0 0 1 -40.867 -702.503 cm BT /F21 6.6311 Tf 40.867 702.503 Td [(920)]TJ/F108 6.6311 Tf 131.912 0 Td [(M.)-250(Jerkovi\242)-250(/)-250(Journal)-250(of)-250(Pure)-250(and)-250(Applied)-250(Algebra)-250(213)-250(\0502009\051)-250(913\025926)]TJ ET 1 0 0 1 507.118 702.503 cm 0 g 0 G 1 0 0 1 -470.237 -23.91 cm 0 g 0 G 1 0 0 1 -36.881 -678.593 cm BT /F123 8.2889 Tf 40.867 678.593 Td [(Lemma)-250(5.6.)]TJ ET 1 0 0 1 86.522 678.593 cm 0 g 0 G 1 0 0 1 -86.522 -678.593 cm BT /F108 8.2889 Tf 90.507 678.593 Td [(For)-340(A)]TJ/F158 9.5641 Tf 22.247 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F108 8.2889 Tf 5.639 0 Td [(the)-250(set)]TJ/F193 8.9265 Tf 26.736 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(A)]TJ/F108 8.2889 Tf 6.092 1.435 Td [(can)-250(be)-250(expressed)-250(as)-250(a)-250(disjoint)-250(union)]TJ ET 1 0 0 1 40.867 678.593 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -678.593 cm BT /F193 8.9265 Tf 64.777 658.332 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(A)]TJ/F158 9.5641 Tf 6.676 1.434 Td [(D)]TJ/F159 9.5641 Tf 14.392 9.086 Td [([)]TJ/F108 4.9734 Tf -3.08 -19.032 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -11.984 -5.739 Td [(B)]TJ/F158 5.7385 Tf 2.86 0 Td [(\023)]TJ/F108 4.9734 Tf 4.476 0 Td [(A)]TJ/F193 8.9265 Tf 9.578 15.685 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.95 Td [(B)]TJ/F156 9.5641 Tf 3.835 -3.95 Td [(:)]TJ/F21 8.2889 Tf 366.951 0 Td [(\0505.2\051)]TJ ET 1 0 0 1 40.867 658.332 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -39.761 cm 0 g 0 G 1 0 0 1 -36.882 -618.571 cm BT /F123 8.2889 Tf 40.867 618.571 Td [(Proof.)]TJ ET 1 0 0 1 65.095 618.571 cm 0 g 0 G 1 0 0 1 -65.095 -618.571 cm BT /F21 8.2889 Tf 69.08 618.571 Td [(One)-427(inclusion)-426(is)-427(clear:)-427(for)]TJ/F108 8.2889 Tf 106.314 0 Td [(B)]TJ/F158 9.5641 Tf 11.179 0 Td [(2)]TJ/F108 8.2889 Tf 12.428 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)]TJ/F108 8.2889 Tf 5.478 0 Td [(B)]TJ/F158 9.5641 Tf 11.179 0 Td [(\023)]TJ/F108 8.2889 Tf 13.872 0 Td [(A)]TJ/F21 8.2889 Tf 5.032 0 Td [(,)-427(we)-427(have)]TJ/F193 8.9265 Tf 42.201 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F158 9.5641 Tf 10.247 -3.472 Td [(\022)]TJ/F193 8.9265 Tf 13.873 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F158 9.5641 Tf 10.247 1.434 Td [(\022)]TJ/F193 8.9265 Tf 13.872 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(A)]TJ/F21 8.2889 Tf 4.02 1.434 Td [(.)-428(On)-426(the)-427(other)-427(hand,)-427(take)]TJ/F108 8.2889 Tf 104.751 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 20.242 0 Td [(D)]TJ ET 1 0 0 1 40.867 618.571 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -618.571 cm BT /F156 9.5641 Tf 40.867 608.111 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.681 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.471 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.079 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.875 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.471 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.876 -2.118 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.36 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.471 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F21 8.2889 Tf 5.998 -2.118 Td [(in)]TJ/F193 8.9265 Tf 9.632 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(A)]TJ/F21 8.2889 Tf 6.142 1.435 Td [(and)-256(define)]TJ/F108 8.2889 Tf 42.415 0 Td [(J)]TJ/F158 9.5641 Tf 5.997 0 Td [(D)-291(f)]TJ/F108 8.2889 Tf 13.784 0 Td [(j)]TJ/F158 9.5641 Tf 5.18 0 Td [(j)]TJ/F108 8.2889 Tf 5.521 0 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.034 1.435 Td [(C)-228(\001)-166(\001)-167(\001)-228(C)]TJ/F108 8.2889 Tf 32.615 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.435 Td [(j)]TJ/F158 9.5641 Tf 4.96 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.245 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.034 1.435 Td [(C)-228(\001)-166(\001)-167(\001)-228(C)]TJ/F108 8.2889 Tf 32.615 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.435 Td [(j)]TJ/F158 9.5641 Tf 2.175 1.435 Td [(g)]TJ/F21 8.2889 Tf 3.539 0 Td [(.)-256(By)]TJ ET 1 0 0 1 417.307 608.111 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -417.307 -608.111 cm BT /F21 8.2889 Tf 417.307 608.111 Td [(Lemma)-256(5.5)]TJ ET 1 0 0 1 459.133 608.111 cm 0 g 0 G 1 0 0 1 -459.133 -608.111 cm BT /F21 8.2889 Tf 461.255 608.111 Td [(it)-256(is)-256(obvious)]TJ ET 1 0 0 1 40.867 608.111 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -608.111 cm BT /F21 8.2889 Tf 40.867 596.289 Td [(that)]TJ/F108 8.2889 Tf 17.896 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 17.317 0 Td [(2)]TJ/F193 8.9265 Tf 9.503 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F21 8.2889 Tf 6.23 -3.472 Td [(for)]TJ/F108 8.2889 Tf 13.254 0 Td [(B)]TJ/F158 9.5641 Tf 8.254 0 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ/F158 9.5641 Tf 7.969 1.435 Td [(n)]TJ/F108 8.2889 Tf 7.096 0 Td [(J)]TJ/F21 8.2889 Tf 5.608 0 Td [(and)-289(that)]TJ/F108 8.2889 Tf 34.541 0 Td [(B)]TJ/F158 9.5641 Tf 8.253 0 Td [(\023)]TJ/F108 8.2889 Tf 10.947 0 Td [(A)]TJ/F21 8.2889 Tf 7.428 0 Td [(\050of)-289(course,)-289(in)-289(the)-289(case)-289(of)]TJ/F108 8.2889 Tf 95.715 0 Td [(J)]TJ/F158 9.5641 Tf 6.699 0 Td [(D)-365(;)]TJ/F21 8.2889 Tf 18.661 0 Td [(we)-289(have)]TJ/F108 8.2889 Tf 34.441 0 Td [(B)]TJ/F158 9.5641 Tf 8.254 0 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ/F21 8.2889 Tf 5.511 1.435 Td [(\051)-289(and)-289(the)-289(equality)-289(stated)-289(in)]TJ ET 1 0 0 1 40.867 596.289 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -596.289 cm BT /F21 8.2889 Tf 40.867 585.828 Td [(the)-265(lemma)-266(holds.)-265(Furthermore,)-266(suppose)-265(that)-265(for)-266(some)]TJ/F108 8.2889 Tf 208.052 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.434 Td [(2)]TJ/F158 9.5641 Tf 6.842 1.434 Td [(2)]TJ/F108 8.2889 Tf 9 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.14 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.434 Td [(2)]TJ/F158 9.5641 Tf 6.842 1.434 Td [(\023)]TJ/F108 8.2889 Tf 10.444 0 Td [(A)]TJ/F21 8.2889 Tf 5.031 0 Td [(,)-265(there)-266(exists)]TJ/F108 8.2889 Tf 50.705 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.813 0 Td [(2)]TJ/F193 8.9265 Tf 9 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 9.5641 Tf 6.132 -2.119 Td [(\134)]TJ/F193 8.9265 Tf 8.645 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(2)]TJ/F21 8.2889 Tf 3.875 -2.119 Td [(.)-266(But,)-265(then)]TJ ET 1 0 0 1 40.867 585.828 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -585.828 cm BT /F21 8.2889 Tf 40.867 575.367 Td [(we)-255(have)]TJ/F108 8.2889 Tf 33.87 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.584 0 Td [(2)]TJ/F193 8.9265 Tf 8.771 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F21 8.2889 Tf 5.986 2.787 Td [(and)]TJ/F108 8.2889 Tf 16.359 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.585 0 Td [(2)]TJ/F193 8.9265 Tf 8.771 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(2)]TJ/F21 8.2889 Tf 3.875 2.788 Td [(,)-255(so)]TJ/F108 8.2889 Tf 14.558 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.585 0 Td [(2)]TJ/F193 8.9265 Tf 8.771 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 9.5641 Tf 6.04 2.788 Td [(\134)]TJ/F193 8.9265 Tf 8.554 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(2)]TJ/F158 9.5641 Tf 6.63 2.788 Td [(D)]TJ/F193 8.9265 Tf 10.215 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [([)]TJ/F108 5.8022 Tf 4.472 0 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F21 8.2889 Tf 3.876 2.788 Td [(,)-255(which)-254(is,)-255(due)-254(to)]TJ/F108 8.2889 Tf 66.304 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.435 Td [(1)]TJ/F158 9.5641 Tf 6.022 1.435 Td [([)]TJ/F108 8.2889 Tf 8.553 0 Td [(B)]TJ/F21 5.8022 Tf 4.767 -1.435 Td [(2)]TJ/F158 9.5641 Tf 6.612 1.435 Td [(\033)]TJ/F108 8.2889 Tf 10.215 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.435 Td [(2)]TJ/F21 8.2889 Tf 3.858 1.435 Td [(,)-255(in)-254(opposition)-255(with)]TJ ET 1 0 0 1 40.867 575.367 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -575.367 cm BT /F108 8.2889 Tf 40.867 564.907 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.487 0 Td [(2)]TJ/F193 8.9265 Tf 8.672 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F21 8.2889 Tf 5.948 2.788 Td [(and)]TJ/F108 8.2889 Tf 16.321 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(2)]TJ/F21 8.2889 Tf 3.875 2.788 Td [(.)-250(We)-250(conclude)-250(that)-250(the)-250(union)-250(on)-250(the)-250(right-hand)-250(side)-250(of)]TJ ET 1 0 0 1 355.427 564.907 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -355.427 -564.907 cm BT /F21 8.2889 Tf 355.427 564.907 Td [(\0505.2\051)]TJ ET 1 0 0 1 373.099 564.907 cm 0 g 0 G 1 0 0 1 -373.099 -564.907 cm BT /F21 8.2889 Tf 375.171 564.907 Td [(must)-250(be)-250(disjoint.)]TJ/F137 7.9701 Tf 76.092 0 Td [(\003)]TJ ET 1 0 0 1 40.867 564.907 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -564.907 cm BT /F108 8.2889 Tf 40.867 539.102 Td [(5.2.)-500(Proof)-250(of)-250(exactness)]TJ ET 1 0 0 1 40.867 539.102 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -539.102 cm BT /F21 8.2889 Tf 52.822 517.268 Td [(We)-573(prove)]TJ ET 1 0 0 1 97.223 517.268 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -97.223 -517.268 cm BT /F21 8.2889 Tf 97.223 517.268 Td [(Theorem)-573(5.1)]TJ ET 1 0 0 1 147.667 517.268 cm 0 g 0 G 1 0 0 1 -147.667 -517.268 cm BT /F21 8.2889 Tf 147.667 517.268 Td [(.)-575(It)-573(is)-572(clear)-573(from)-573(the)-573(definition)-573(that)]TJ/F158 9.5641 Tf 154.162 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)-995(D)-995(T)]TJ/F156 9.5641 Tf 32.866 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F158 6.6948 Tf 3.185 3.471 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 8.2889 Tf 8.263 -3.471 Td [(is)-573(injective.)-575(Therefore,)-573(as)-573(the)-572(first)]TJ ET 1 0 0 1 40.867 517.268 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -517.268 cm BT /F21 8.2889 Tf 40.867 506.807 Td [(nontrivial)-472(step)-472(we)-471(prove)-472(that)]TJ/F108 8.2889 Tf 122.022 0 Td [(Im)]TJ/F156 9.5641 Tf 9.656 0 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(T)]TJ/F156 9.5641 Tf 3.184 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F158 6.6948 Tf 3.185 3.472 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 9.5641 Tf 3.515 -3.472 Td [(/)]TJ/F158 9.5641 Tf 10.939 0 Td [(D)]TJ/F108 8.2889 Tf 14.832 0 Td [(Ker)]TJ/F156 9.5641 Tf 13.125 0 Td [(.')]TJ/F21 5.8022 Tf 8.943 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(.)-473(Recall)-472(that)]TJ/F193 8.9265 Tf 51.895 0 Td [(B)]TJ/F158 9.5641 Tf 15.263 0 Td [(D)]TJ/F193 8.9265 Tf 14.832 0 Td [(B)]TJ/F158 6.6948 Tf 7.319 -1.435 Td [(;)]TJ/F21 8.2889 Tf 8.132 1.435 Td [(and)-472(let)]TJ/F108 8.2889 Tf 32.001 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(/v)]TJ/F158 9.5641 Tf 26.242 0 Td [(2)]TJ/F108 8.2889 Tf 13.388 0 Td [(Ker)]TJ/F156 9.5641 Tf 13.125 0 Td [(.')]TJ/F21 5.8022 Tf 8.942 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F21 8.2889 Tf 7.478 0 Td [(for)]TJ/F108 8.2889 Tf 14.77 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 21.201 0 Td [(D)]TJ ET 1 0 0 1 40.867 506.807 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -506.807 cm BT /F156 9.5641 Tf 40.867 496.346 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.681 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.079 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.875 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.876 -2.119 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.36 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(0)]TJ/F158 9.5641 Tf 6.532 -2.119 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F158 6.6948 Tf 7.32 -1.435 Td [(;)]TJ/F21 8.2889 Tf 4.22 1.435 Td [(.)-250(From)]TJ ET 1 0 0 1 40.867 496.346 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -496.346 cm BT /F156 9.5641 Tf 64.777 477.098 Td [(')]TJ/F21 5.8022 Tf 5.375 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)-40(/)]TJ/F158 9.5641 Tf 25.094 0 Td [(D)]TJ/F159 9.5641 Tf 16.371 9.086 Td [(X)]TJ/F108 5.8022 Tf -6.255 -19.689 Td [(I)]TJ/F21 4.9734 Tf 1.747 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F21 4.9734 Tf 4.026 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.091 10.603 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F108 5.8022 Tf 18.488 -1.434 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 9.5641 Tf 6.533 2.787 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ ET 1 0 0 1 40.867 477.098 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -477.098 cm BT /F21 8.2889 Tf 40.867 450.249 Td [(by)-250(using)-249(the)-250(fact)-249(that)-250(for)]TJ/F108 8.2889 Tf 95.384 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 15.899 0 Td [(which)-250(satisfies)-249(the)-250(difference)-249(conditions)-250(and)-249(does)-250(not)-250(satisfy)-249(the)-250(initial)-249(conditions)-250(for)]TJ/F108 8.2889 Tf 327.191 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ ET 1 0 0 1 40.867 450.249 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -450.249 cm BT /F21 8.2889 Tf 40.867 439.788 Td [(the)-250(assertion)]TJ/F108 8.2889 Tf 51.366 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)-100(/v)]TJ/F156 6.4438 Tf 18.488 -1.435 Td [(\003)]TJ/F158 9.5641 Tf 8.284 1.435 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)-250(holds,)-250(one)-250(gets)]TJ ET 1 0 0 1 40.867 439.788 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -439.788 cm BT /F108 8.2889 Tf 64.777 421.694 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F158 6.6948 Tf 7.319 -1.435 Td [(;)]TJ/F158 9.5641 Tf 6.346 1.435 Td [(n)]TJ/F159 9.5641 Tf 13.917 9.086 Td [([)]TJ/F108 5.8022 Tf -7.153 -19.69 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F21 4.9734 Tf 4.027 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F193 8.9265 Tf 4.091 10.604 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 9.5641 Tf 6.533 2.788 Td [(D)]TJ/F193 8.9265 Tf 10.116 0 Td [(B)]TJ/F158 6.6948 Tf 7.32 -1.435 Td [(;)]TJ/F158 9.5641 Tf 6.346 1.435 Td [(n)]TJ/F159 9.5641 Tf 11.637 9.086 Td [([)]TJ/F108 5.8022 Tf -4.873 -19.69 Td [(C)]TJ/F158 6.6948 Tf 3.942 0 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F156 6.6948 Tf 4.027 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F193 8.9265 Tf 4.091 10.604 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(C)]TJ/F158 9.5641 Tf 7.097 1.435 Td [(D)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F158 6.6948 Tf 7.891 3.95 Td [(;)]TJ/F156 9.5641 Tf 4.22 -3.95 Td [(:)]TJ ET 1 0 0 1 40.867 421.694 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -421.694 cm BT /F21 8.2889 Tf 40.867 394.844 Td [(By)-250(using)]TJ ET 1 0 0 1 75.24 394.844 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -75.24 -394.844 cm BT /F21 8.2889 Tf 75.24 394.844 Td [(Lemma)-250(5.5)]TJ ET 1 0 0 1 117.016 394.844 cm 0 g 0 G 1 0 0 1 -117.016 -394.844 cm BT /F21 8.2889 Tf 119.088 394.844 Td [(we)-250(conclude)-250(that)-250(for)]TJ/F108 8.2889 Tf 80.543 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ ET 1 0 0 1 40.867 394.844 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -394.844 cm BT /F108 8.2889 Tf 64.777 378.12 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.435 Td [(j)]TJ/F158 9.5641 Tf 4.832 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-166(\001)-223(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.435 Td [(j)]TJ/F156 9.5641 Tf 2.175 1.435 Td [(;)]TJ/F108 8.2889 Tf 13.049 0 Td [(j)]TJ/F158 9.5641 Tf 5.052 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(J)]TJ/F158 6.6948 Tf 2.462 -1.435 Td [(;)]TJ/F158 9.5641 Tf 6.877 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ ET 1 0 0 1 40.867 378.12 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -378.12 cm BT /F21 8.2889 Tf 40.867 361.395 Td [(holds.)-238(Since)-237(every)]TJ/F108 8.2889 Tf 70.037 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.435 Td [(i)]TJ/F156 9.5641 Tf 2.158 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(i)]TJ/F158 9.5641 Tf 5.028 0 Td [(2)-278(f)]TJ/F21 8.2889 Tf 12.211 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F158 9.5641 Tf 28.475 0 Td [(\000)]TJ/F21 8.2889 Tf 9.267 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)-189(n)]TJ/F108 8.2889 Tf 11.791 0 Td [(J)]TJ/F158 6.6948 Tf 2.462 -1.435 Td [(;)]TJ/F21 8.2889 Tf 4.22 1.435 Td [(,)-238(equals)-237(zero,)-238(the)-237(above)-238(equalities)-237(imply)-238(the)-237(equality)-238(between)]TJ/F108 8.2889 Tf 235.869 0 Td [(all)]TJ/F21 8.2889 Tf 10.929 0 Td [(corresponding)]TJ ET 1 0 0 1 40.867 361.395 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -361.395 cm BT /F21 8.2889 Tf 40.867 350.934 Td [(left-hand)-250(and)-250(right-hand)-250(partial)-250(sums:)]TJ ET 1 0 0 1 40.867 350.934 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -350.934 cm BT /F108 8.2889 Tf 64.777 334.21 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.814 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F108 5.8022 Tf 4.311 -1.435 Td [(i)]TJ/F156 9.5641 Tf 2.157 1.435 Td [(;)]TJ/F108 8.2889 Tf 13.049 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(2)-278(f)]TJ/F21 8.2889 Tf 12.211 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F158 9.5641 Tf 28.794 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(:)]TJ ET 1 0 0 1 40.867 334.21 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -334.21 cm BT /F21 8.2889 Tf 40.867 317.485 Td [(We)-250(now)-250(have)]TJ/F108 8.2889 Tf 54.225 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.434 Td [(i)]TJ/F158 9.5641 Tf 4.814 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.434 Td [(i)]TJ/F156 9.5641 Tf 2.158 1.434 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(i)]TJ/F158 9.5641 Tf 5.028 0 Td [(2)-278(f)]TJ/F21 8.2889 Tf 12.211 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F158 9.5641 Tf 28.793 0 Td [(\000)]TJ/F21 8.2889 Tf 9.586 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)]TJ/F21 8.2889 Tf 3.539 0 Td [(,)-250(and)]TJ ET 1 0 0 1 40.867 317.485 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -317.485 cm BT /F108 8.2889 Tf 64.777 300.761 Td [(Ker)]TJ/F156 9.5641 Tf 13.125 0 Td [(.')]TJ/F21 5.8022 Tf 8.943 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(span)]TJ/F158 9.5641 Tf 17.97 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F158 9.5641 Tf 21.526 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(W)]TJ/F158 9.5641 Tf 11.903 0 Td [(j)]TJ/F108 8.2889 Tf 5.392 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.681 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.557 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F158 9.5641 Tf 3.876 -2.597 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(:)]TJ ET 1 0 0 1 40.867 300.761 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -300.761 cm BT /F21 8.2889 Tf 52.822 283.808 Td [(On)-250(the)-250(other)-250(hand,)-250(take)]TJ/F108 8.2889 Tf 91.931 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F21 5.8022 Tf 9.306 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/v)]TJ/F108 5.8022 Tf 8.225 -1.434 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.393 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F21 8.2889 Tf 5.948 2.827 Td [(where)]TJ ET 1 0 0 1 40.867 283.808 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -283.808 cm BT /F108 8.2889 Tf 64.777 264.843 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F21 5.8022 Tf 9.306 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(b)]TJ/F156 5.7385 Tf 3.15 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.557 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(b)]TJ/F21 4.9734 Tf 3.15 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.876 -2.597 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(b)]TJ/F21 4.9734 Tf 3.15 -1.353 Td [(0)]TJ/F158 9.5641 Tf 6.533 -2.597 Td [(2)]TJ/F193 8.9265 Tf 8.672 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ ET 1 0 0 1 40.867 264.843 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -264.843 cm BT /F21 8.2889 Tf 40.867 248.118 Td [(and)-250(calculate)-250(using)]TJ ET 1 0 0 1 115.93 248.118 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -115.93 -248.118 cm BT /F21 8.2889 Tf 115.93 248.118 Td [(\0504.2\051)]TJ ET 1 0 0 1 133.602 248.118 cm 0 g 0 G 1 0 0 1 -133.602 -248.118 cm BT /F21 8.2889 Tf 135.674 248.118 Td [(and)]TJ ET 1 0 0 1 151.995 248.118 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -151.995 -248.118 cm BT /F21 8.2889 Tf 151.995 248.118 Td [(\0504.3\051)]TJ ET 1 0 0 1 169.667 248.118 cm 0 g 0 G 1 0 0 1 -169.667 -248.118 cm BT /F21 8.2889 Tf 169.667 248.118 Td [(:)]TJ ET 1 0 0 1 40.867 248.118 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -248.118 cm BT /F158 9.5641 Tf 64.777 231.394 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F158 6.6948 Tf 3.184 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 9.5641 Tf 3.515 -3.95 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F21 5.8022 Tf 9.306 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/v)]TJ/F108 5.8022 Tf 8.225 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.827 Td [(/)]TJ/F158 9.5641 Tf 7.054 0 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F158 6.6948 Tf 10.262 4.075 Td [(\000)]TJ/F21 5.8022 Tf -0.956 -6.768 Td [(1)]TJ/F156 9.5641 Tf 6.676 2.693 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.557 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.875 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.875 -2.597 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F158 9.5641 Tf 3.876 -2.597 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(w)]TJ/F158 9.5641 Tf 7.316 0 Td [(U)]TJ/F158 6.6948 Tf 3.185 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 9.5641 Tf 3.515 -3.95 Td [(.v)]TJ/F108 5.8022 Tf 8.225 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.827 Td [(/)]TJ ET 1 0 0 1 40.867 231.394 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -231.394 cm BT /F158 9.5641 Tf 161.375 215.704 Td [(D)]TJ/F108 8.2889 Tf 10.947 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F158 6.6948 Tf 10.262 4.075 Td [(\000)]TJ/F21 5.8022 Tf -0.956 -6.768 Td [(1)]TJ/F156 9.5641 Tf 6.676 2.693 Td [(/)]TJ/F108 8.2889 Tf 3.568 0 Td [(x)]TJ/F156 6.6948 Tf 4.069 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.828 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.557 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.875 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.875 -2.597 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 9.5641 Tf 3.876 -2.597 Td [(v)-40(:)]TJ ET 1 0 0 1 40.867 215.704 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -215.704 cm BT /F21 8.2889 Tf 40.867 198.979 Td [(If)-250(vector)]TJ/F108 8.2889 Tf 33.486 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F21 5.8022 Tf 9.306 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/v)]TJ/F108 5.8022 Tf 8.225 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F21 8.2889 Tf 5.949 2.827 Td [(satisfies)-250(the)-250(initial)-250(conditions)-250(for)]TJ/F108 8.2889 Tf 126.719 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F21 8.2889 Tf 3.876 2.827 Td [(,)-250(then)]TJ ET 1 0 0 1 40.867 198.979 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -198.979 cm BT /F108 8.2889 Tf 64.777 180.014 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F158 6.6948 Tf 10.262 4.075 Td [(\000)]TJ/F21 5.8022 Tf -0.956 -6.768 Td [(1)]TJ/F156 9.5641 Tf 6.676 2.693 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.557 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.875 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.875 -2.597 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 9.5641 Tf 3.876 -2.597 Td [(v)]TJ ET 1 0 0 1 40.867 180.014 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -180.014 cm BT /F21 8.2889 Tf 40.867 163.289 Td [(satisfies)-250(the)-250(difference)-250(conditions.)-250(It)-250(is)-250(therefore)-250(enough)-250(to)-250(check:)]TJ ET 1 0 0 1 40.867 163.289 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -163.289 cm BT /F108 8.2889 Tf 64.777 146.565 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.435 Td [(\051)]TJ/F108 8.2889 Tf 12.632 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ ET 1 0 0 1 40.867 146.565 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -146.565 cm BT /F108 8.2889 Tf 64.777 133.115 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.434 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.434 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 9.5641 Tf 5.757 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F158 9.5641 Tf 6.515 1.434 Td [(\051)]TJ/F108 8.2889 Tf 12.632 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 6.6948 Tf 3.134 0 Td [(\000)]TJ/F21 5.8022 Tf 5.221 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.586 0 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.982 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.586 0 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.434 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.434 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ ET 1 0 0 1 40.867 133.115 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -133.115 cm BT /F156 9.5641 Tf 64.777 119.666 Td [(:)-167(:)-166(:)]TJ ET 1 0 0 1 40.867 119.666 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -119.666 cm BT /F108 8.2889 Tf 64.777 106.216 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(b)]TJ/F156 6.6948 Tf 4.501 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 5.757 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.982 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(2)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\051)]TJ/F108 8.2889 Tf 12.632 0 Td [(k)]TJ/F156 6.6948 Tf 4.311 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(b)]TJ/F21 5.8022 Tf 4.501 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(b)]TJ/F156 6.6948 Tf 4.501 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F156 9.5641 Tf 4.311 0 Td [(:)]TJ ET 1 0 0 1 40.867 106.216 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -106.216 cm BT /F21 8.2889 Tf 40.867 89.492 Td [(Also,)-250(the)-250(image)-250(of)]TJ/F108 8.2889 Tf 69.825 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)]TJ/F21 5.8022 Tf 9.305 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/v)]TJ/F108 5.8022 Tf 8.225 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.393 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 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8.2889 Tf 9.585 0 Td [(1)]TJ/F108 8.2889 Tf -128.261 -13.449 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.435 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F108 5.8022 Tf 4.311 -1.435 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf -129.856 -12.015 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 0 -13.449 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F156 6.6948 Tf 4.418 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.134 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)]TJ/F108 8.2889 Tf 22.344 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 307.351 61.659 Td [(\0505.1\051)]TJ ET 1 0 0 1 37.134 600.733 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -600.733 cm BT /F21 8.2889 Tf 37.134 522.849 Td [(\050in)-327(the)-327(case)-328(of)]TJ/F108 8.2889 Tf 57.682 0 Td [(A)]TJ/F158 9.5641 Tf 9.329 0 Td [(D)-449(;)]TJ/F21 8.2889 Tf 19.789 0 Td [(these)-327(are)-327(just)]TJ ET 1 0 0 1 178.961 522.849 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -178.961 -522.849 cm BT /F21 8.2889 Tf 178.961 522.849 Td [(\0503.2\051)]TJ ET 1 0 0 1 196.633 522.849 cm 0 g 0 G 1 0 0 1 -196.633 -522.849 cm BT /F21 8.2889 Tf 196.633 522.849 Td [(\051.)-328(It)-327(is)-327(obvious)-327(that)-328(the)-327(initial)-327(conditions)-327(for)]TJ/F108 8.2889 Tf 172.413 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(B)]TJ/F21 8.2889 Tf 6.547 1.435 Td [(follow)-327(from)-327(those)-328(for)]TJ/F108 8.2889 Tf 85.358 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(A)]TJ/F21 8.2889 Tf 6.732 1.435 Td [(if)-327(we)]TJ ET 1 0 0 1 37.134 522.849 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -522.849 cm BT /F21 8.2889 Tf 37.134 512.388 Td [(strengthen)-250(every)]TJ/F156 9.5641 Tf 66.858 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(j)]TJ/F158 9.5641 Tf 4.521 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(th)-250(inequality)-250(in)]TJ ET 1 0 0 1 190.474 512.388 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -190.474 -512.388 cm BT /F21 8.2889 Tf 190.474 512.388 Td [(\0505.1\051)]TJ ET 1 0 0 1 208.146 512.388 cm 0 g 0 G 1 0 0 1 -208.146 -512.388 cm BT /F21 8.2889 Tf 210.218 512.388 Td [(for)-250(each)]TJ/F108 8.2889 Tf 32.65 0 Td [(j)]TJ/F158 9.5641 Tf 5.052 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(B)]TJ/F158 9.5641 Tf 6.892 0 Td [(n)]TJ/F108 8.2889 Tf 6.764 0 Td [(A)]TJ/F21 8.2889 Tf 5.031 0 Td [(:)]TJ ET 1 0 0 1 37.134 512.388 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -512.388 cm BT /F108 8.2889 Tf 61.044 495.026 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.435 Td [(j)]TJ/F158 9.5641 Tf 4.832 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.461 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.435 Td [(j)]TJ/F158 9.5641 Tf 4.3 1.435 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(:)]TJ ET 1 0 0 1 37.134 495.026 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -495.026 cm BT /F21 8.2889 Tf 37.134 477.663 Td [(Hence,)-250(for)-250(every)]TJ/F108 8.2889 Tf 64.064 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 8.2889 Tf 5.906 1.435 Td [(also)]TJ/F108 8.2889 Tf 17.241 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(A)]TJ/F21 8.2889 Tf 6.093 1.435 Td [(holds.)]TJ/F137 7.9701 Tf 35.211 0 Td [(\003)]TJ ET 1 0 0 1 37.134 477.663 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -22.526 cm 0 g 0 G 1 0 0 1 -33.149 -455.137 cm BT /F123 8.2889 Tf 37.134 455.137 Td [(Lemma)-250(5.4.)]TJ ET 1 0 0 1 82.789 455.137 cm 0 g 0 G 1 0 0 1 -82.789 -455.137 cm BT /F108 8.2889 Tf 86.774 455.137 Td [(Let)-335(B)]TJ/F21 5.8022 Tf 18.389 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(B)]TJ/F21 5.8022 Tf 4.766 -1.435 Td [(2)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F108 8.2889 Tf 3.568 0 Td [(.)-250(Then)]TJ ET 1 0 0 1 37.134 455.137 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -455.137 cm BT /F193 8.9265 Tf 61.044 437.774 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 9.5641 Tf 6.001 2.788 Td [(\134)]TJ/F193 8.9265 Tf 8.514 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F158 9.5641 Tf 6.533 2.788 Td [(D)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [([)]TJ/F108 5.8022 Tf 4.472 0 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.875 2.788 Td [(:)]TJ ET 1 0 0 1 37.134 437.774 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -25.19 cm 0 g 0 G 1 0 0 1 -33.149 -412.584 cm BT /F123 8.2889 Tf 37.134 412.584 Td [(Proof.)]TJ ET 1 0 0 1 61.362 412.584 cm 0 g 0 G 1 0 0 1 -61.362 -412.584 cm BT /F21 8.2889 Tf 65.347 412.584 Td [(From)]TJ ET 1 0 0 1 87.42 412.584 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -87.42 -412.584 cm BT /F21 8.2889 Tf 87.42 412.584 Td [(Lemma)-250(5.3)]TJ ET 1 0 0 1 129.196 412.584 cm 0 g 0 G 1 0 0 1 -129.196 -412.584 cm BT /F21 8.2889 Tf 131.268 412.584 Td [(we)-250(have)]TJ ET 1 0 0 1 37.134 412.584 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -412.584 cm BT /F193 8.9265 Tf 61.044 395.222 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [([)]TJ/F108 5.8022 Tf 4.472 0 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F158 9.5641 Tf 6.533 2.788 Td [(\022)]TJ/F193 8.9265 Tf 10.116 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ ET 1 0 0 1 37.134 395.222 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -395.222 cm BT /F193 8.9265 Tf 61.044 381.772 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [([)]TJ/F108 5.8022 Tf 4.472 0 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F158 9.5641 Tf 6.533 2.788 Td [(\022)]TJ/F193 8.9265 Tf 10.116 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(;)]TJ ET 1 0 0 1 37.134 381.772 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -381.772 cm BT /F21 8.2889 Tf 37.134 364.409 Td [(so)]TJ/F193 8.9265 Tf 10.36 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [([)]TJ/F108 5.8022 Tf 4.472 0 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F158 9.5641 Tf 6.532 2.787 Td [(\022)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 9.5641 Tf 5.666 2.787 Td [(\134)]TJ/F193 8.9265 Tf 8.18 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F21 8.2889 Tf 5.839 2.787 Td [(holds.)-237(On)-237(the)-237(other)-237(hand,)-236(a)-237(monomial)]TJ/F108 8.2889 Tf 145.081 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.487 0 Td [(2)]TJ/F193 8.9265 Tf 8.672 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 9.5641 Tf 5.666 2.787 Td [(\134)]TJ/F193 8.9265 Tf 8.18 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F21 8.2889 Tf 5.839 2.787 Td [(satisfies)-237(the)-237(initial)-237(conditions)-236(both)-237(for)]TJ/F108 8.2889 Tf 145.753 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(1)]TJ ET 1 0 0 1 37.134 364.409 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -364.409 cm BT /F21 8.2889 Tf 37.134 353.948 Td [(and)]TJ/F108 8.2889 Tf 16.559 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.434 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(2)]TJ/F21 8.2889 Tf 3.876 2.788 Td [(,)-279(which)-279(means)-278(that)-279(it)-279(always)-279(satisfies)-279(the)-278(stronger)-279(inequality)-279(out)-279(of)-279(two)-278(equally)-279(indexed)-279(ones,)-279(which)-279(implies)-278(that)]TJ ET 1 0 0 1 37.134 353.948 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -353.948 cm BT /F108 8.2889 Tf 37.134 343.488 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(2)]TJ/F193 8.9265 Tf 8.673 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(B)]TJ/F21 4.9734 Tf 3.336 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [([)]TJ/F108 5.8022 Tf 4.472 0 Td [(B)]TJ/F21 4.9734 Tf 3.337 -1.353 Td [(2)]TJ/F21 8.2889 Tf 3.875 2.788 Td [(.)]TJ/F137 7.9701 Tf 14.382 0 Td [(\003)]TJ ET 1 0 0 1 37.134 343.488 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -343.488 cm BT /F21 8.2889 Tf 49.089 325.2 Td [(For)]TJ/F108 8.2889 Tf 14.389 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(let)-250(us)-250(define)]TJ ET 1 0 0 1 37.134 325.2 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -325.2 cm BT /F193 8.9265 Tf 61.044 306.488 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.95 Td [(B)]TJ/F158 9.5641 Tf 6.491 -3.95 Td [(D)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(B)]TJ/F158 9.5641 Tf 5.959 1.434 Td [(n)]TJ/F159 9.5641 Tf 11.299 9.086 Td [([)]TJ/F108 4.9734 Tf -3.339 -19.032 Td [(C)]TJ/F158 5.7385 Tf 3.379 0 Td [(2)]TJ/F108 4.9734 Tf 3.61 0 Td [(D)]TJ/F156 5.7385 Tf 3.451 0 Td [(.)]TJ/F108 4.9734 Tf 2.141 0 Td [(K)]TJ/F156 5.7385 Tf 3.692 0 Td [(/)]TJ/F108 4.9734 Tf -12.424 -5.738 Td [(C)]TJ/F158 5.7385 Tf 3.38 0 Td [(\033)]TJ/F108 4.9734 Tf 4.476 0 Td [(B)]TJ/F193 8.9265 Tf 9.498 15.684 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(C)]TJ/F156 9.5641 Tf 4.441 1.434 Td [(:)]TJ ET 1 0 0 1 37.134 306.488 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -306.488 cm BT /F21 8.2889 Tf 49.089 274.611 Td [(The)-250(next)-250(lemma)-250(describes)-250(the)-250(above)-250(defined)-250(set)-250(more)-250(explicitly:)]TJ ET 1 0 0 1 37.134 274.611 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -22.988 cm 0 g 0 G 1 0 0 1 -33.149 -251.623 cm BT /F123 8.2889 Tf 37.134 251.623 Td [(Lemma)-250(5.5.)]TJ ET 1 0 0 1 82.789 251.623 cm 0 g 0 G 1 0 0 1 -82.789 -251.623 cm BT /F108 8.2889 Tf 86.774 251.623 Td [(For)-389(B)]TJ/F158 9.5641 Tf 23.426 0 Td [(2)]TJ/F108 8.2889 Tf 9.712 0 Td [(D)]TJ/F156 9.5641 Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F108 8.2889 Tf 6.045 0 Td [(and)-299(J)]TJ/F108 5.8022 Tf 18.757 -1.435 Td [(B)]TJ/F158 9.5641 Tf 7.53 1.435 Td [(D)]TJ/F108 8.2889 Tf 11.155 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ/F158 9.5641 Tf 8.052 1.435 Td [(n)]TJ/F108 8.2889 Tf 7.18 0 Td [(B,)-299(the)-299(set)]TJ/F193 8.9265 Tf 36.671 0 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.472 Td [(B)]TJ/F108 8.2889 Tf 6.312 -3.472 Td [(consists)-299(of)-299(all)-299(x)]TJ/F156 9.5641 Tf 55.756 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 17.526 0 Td [(D)]TJ/F156 9.5641 Tf 11.155 0 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.673 0 Td [(x)]TJ/F156 6.6948 Tf 4.069 -1.435 Td [(\015)]TJ/F156 5.7385 Tf 3.361 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F156 5.7385 Tf 3.093 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 5.47 -2.079 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 12.672 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(2)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.472 Td [(a)]TJ/F21 4.9734 Tf 3.093 -1.353 Td [(1)]TJ/F108 8.2889 Tf 3.875 -2.119 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F21 4.9734 Tf 3.361 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.472 Td [(a)]TJ/F21 4.9734 Tf 3.092 -1.353 Td [(0)]TJ/F158 9.5641 Tf 7.572 -2.119 Td [(2)]TJ/F193 8.9265 Tf 9.711 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(B)]TJ ET 1 0 0 1 37.134 251.623 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -251.623 cm BT /F108 8.2889 Tf 37.134 241.162 Td [(such)-250(that)]TJ ET 1 0 0 1 37.134 241.162 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -241.162 cm BT /F108 8.2889 Tf 61.044 223.799 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.434 Td [(j)]TJ/F158 9.5641 Tf 4.832 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.434 Td [(j)]TJ/F156 9.5641 Tf 2.175 1.434 Td [(;)]TJ/F108 8.2889 Tf 13.049 0 Td [(j)]TJ/F158 9.5641 Tf 5.052 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(J)]TJ/F108 5.8022 Tf 2.461 -1.434 Td [(B)]TJ/F156 9.5641 Tf 3.835 1.434 Td [(:)]TJ ET 1 0 0 1 37.134 223.799 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -25.189 cm 0 g 0 G 1 0 0 1 -33.149 -198.61 cm BT /F123 8.2889 Tf 37.134 198.61 Td [(Proof.)]TJ ET 1 0 0 1 61.362 198.61 cm 0 g 0 G 1 0 0 1 -61.362 -198.61 cm BT /F21 8.2889 Tf 65.347 198.61 Td [(Note)-250(that)]TJ/F108 8.2889 Tf 37.905 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ/F21 8.2889 Tf 7.583 1.435 Td [(is)-250(the)-250(largest)-250(set)-250(in)]TJ/F108 8.2889 Tf 73.489 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(and)-250(that)-250(for)]TJ/F108 8.2889 Tf 46.824 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ/F21 8.2889 Tf 7.584 1.435 Td [(we)-250(have)]TJ/F108 8.2889 Tf 33.793 0 Td [(J)]TJ/F108 5.8022 Tf 2.462 -1.435 Td [(B)]TJ/F158 9.5641 Tf 6.491 1.435 Td [(D)-278(;)]TJ/F21 8.2889 Tf 15.434 0 Td [(.)-250(In)-250(this)-250(case)-250(the)-250(lemma)-250(trivially)-250(asserts)]TJ ET 1 0 0 1 37.134 198.61 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -198.61 cm BT /F193 8.9265 Tf 61.044 181.247 Td [(B)]TJ/F108 5.8022 Tf 7.891 3.95 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(m)]TJ/F158 9.5641 Tf 7.95 -2.946 Td [(D)]TJ/F193 8.9265 Tf 10.116 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.005 Td [(m)]TJ/F156 9.5641 Tf 5.294 2.439 Td [(;)]TJ ET 1 0 0 1 37.134 181.247 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -181.247 cm BT /F21 8.2889 Tf 37.134 163.884 Td [(which)-372(is)-372(an)-373(obvious)-372(consequence)-372(of)-372(the)-372(definition)-373(o)1(f)]TJ/F193 8.9265 Tf 210.792 0 Td 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[(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 8.2889 Tf 6.382 0 Td [(and)-250(define)]TJ/F108 8.2889 Tf 42.315 0 Td [(m)]TJ/F158 9.5641 Tf 9.818 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(])]TJ/F158 9.5641 Tf 4.256 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F158 9.5641 Tf 28.793 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F158 9.5641 Tf 7.456 0 Td [(j)]TJ/F108 8.2889 Tf 5.392 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.434 Td [(i)]TJ/F158 9.5641 Tf 4.814 1.434 Td [(6)1(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)]TJ/F21 8.2889 Tf 3.539 0 Td [(.)-250(Denote)]TJ ET 1 0 0 1 40.867 680.585 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -680.585 cm BT /F108 8.2889 Tf 64.777 662.751 Td [(W)]TJ/F158 9.5641 Tf 11.903 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.828 Td [(;)-1900(v)]TJ/F158 9.5641 Tf 29.034 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.828 Td [(;)]TJ/F193 8.9265 Tf 21.338 0 Td [(B)]TJ/F158 9.5641 Tf 10.547 0 Td [(D)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F108 5.8022 Tf 7.32 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.828 Td [(:)]TJ ET 1 0 0 1 40.867 662.751 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -662.751 cm BT /F21 8.2889 Tf 40.867 644.916 Td [(For)]TJ/F108 8.2889 Tf 14.389 0 Td [(t)]TJ/F158 9.5641 Tf 6.3 0 Td [(2)-278(f)]TJ/F21 8.2889 Tf 12.211 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(m)]TJ/F158 9.5641 Tf 9.287 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)]TJ/F21 8.2889 Tf 5.611 0 Td [(define)]TJ ET 1 0 0 1 40.867 644.916 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -644.916 cm BT /F108 8.2889 Tf 64.777 627.081 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.434 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)-278(ff)]TJ/F108 8.2889 Tf 17.194 0 Td [(i)]TJ/F21 5.8022 Tf 2.371 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.434 Td [(t)]TJ/F158 9.5641 Tf 3.048 1.434 Td [(g)-278(j)]TJ/F21 8.2889 Tf 11.587 0 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(i)]TJ/F21 5.8022 Tf 2.371 -1.434 Td [(0)]TJ/F156 9.5641 Tf 6.514 1.434 Td [(<)]TJ/F158 9.5641 Tf 10.116 0 Td [(\001)-167(\001)-166(\001)]TJ/F156 9.5641 Tf 13.822 0 Td [(<)]TJ/F108 8.2889 Tf 10.116 0 Td [(i)]TJ/F108 5.8022 Tf 2.371 -1.434 Td [(t)]TJ/F158 9.5641 Tf 5.705 1.434 Td [(\024)]TJ/F156 9.5641 Tf 9.868 0 Td [(`)]TJ/F158 9.5641 Tf 6.601 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)-250(such)-250(that)]TJ/F108 8.2889 Tf 44.188 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.434 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.353 Td [(j)]TJ/F158 9.5641 Tf 5.09 2.787 Td [(6)1(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(j)]TJ/F158 9.5641 Tf 5.052 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(t)]TJ/F158 9.5641 Tf 3.643 0 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(:)]TJ ET 1 0 0 1 40.867 627.081 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -627.081 cm BT /F21 8.2889 Tf 40.867 608.25 Td [(Let)]TJ/F108 8.2889 Tf 13.66 0 Td [(D)]TJ/F21 5.8022 Tf 5.752 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F158 9.5641 Tf 6.225 0 Td [(V)56(D)-278(f;g)]TJ/F21 8.2889 Tf 26.682 0 Td [(and)-250(denote)-250(by)]TJ/F108 8.2889 Tf 56.273 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(the)-250(union)-250(of)-250(all)]TJ/F108 8.2889 Tf 59.688 0 Td [(D)]TJ/F21 5.8022 Tf 5.753 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/;)]TJ/F108 8.2889 Tf 8.327 0 Td [(D)]TJ/F21 5.8022 Tf 5.753 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 25.759 0 Td [(D)]TJ/F108 5.8022 Tf 5.753 -1.435 Td [(m)]TJ/F156 9.5641 Tf 5.511 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(\050note)-250(that)]TJ/F108 8.2889 Tf 39.886 0 Td [(D)]TJ/F108 5.8022 Tf 5.753 -1.435 Td [(m)]TJ/F156 9.5641 Tf 5.511 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)-278(f)]TJ/F108 8.2889 Tf 13.655 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(m)]TJ/F158 9.5641 Tf 5.511 1.435 Td [(g)]TJ/F21 8.2889 Tf 5.611 0 Td [(is)-250(monadic\051.)]TJ ET 1 0 0 1 40.867 608.25 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -608.25 cm BT /F21 8.2889 Tf 52.822 597.191 Td [(For)]TJ/F108 8.2889 Tf 14.389 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(D)-278(f)]TJ/F108 8.2889 Tf 13.656 0 Td [(i)]TJ/F21 5.8022 Tf 2.37 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(t)]TJ/F158 9.5641 Tf 3.048 1.435 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.868 0 Td [(D)]TJ/F108 5.8022 Tf 5.753 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)]TJ/F108 8.2889 Tf 4.011 0 Td [(t)]TJ/F158 9.5641 Tf 6.3 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(m)]TJ/F158 9.5641 Tf 9.287 0 Td [(\000)]TJ/F21 8.2889 Tf 9.585 0 Td [(1,)-250(we)-250(introduce)-250(the)-250(following)-250(notation:)]TJ ET 1 0 0 1 40.867 597.191 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -597.191 cm BT /F108 8.2889 Tf 64.777 579.356 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 6.533 2.788 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(W)]TJ/F158 6.6948 Tf 8.131 -1.435 Td [(f)]TJ/F108 5.8022 Tf 2.477 0 Td [(i)]TJ/F21 4.9734 Tf 1.66 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [(g)]TJ/F158 9.5641 Tf 5.632 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.352 Td [(i)]TJ/F21 4.9734 Tf 1.422 -1.705 Td [(0)]TJ/F158 6.6948 Tf 3.876 3.057 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.353 Td [(i)]TJ/F21 4.9734 Tf 1.423 -1.705 Td [(0)]TJ/F158 5.7385 Tf 3.377 1.705 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.352 Td [(i)]TJ 1.422 -1.191 Td [(t)]TJ/F158 6.6948 Tf 3.182 2.543 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.353 Td [(i)]TJ 1.422 -1.19 Td [(t)]TJ/F158 5.7385 Tf 2.684 1.19 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(;)]TJ ET 1 0 0 1 40.867 579.356 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -579.356 cm BT /F21 8.2889 Tf 40.867 560.087 Td [(similarly)-250(also)-250(for)-250(highest)-250(weight)-250(vector)]TJ/F156 9.5641 Tf 150.765 0 Td [(v)]TJ/F158 6.6948 Tf 4.658 -1.434 Td [(f)]TJ/F108 5.8022 Tf 2.477 0 Td [(i)]TJ/F21 4.9734 Tf 1.659 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.005 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.005 Td [(g)]TJ/F21 8.2889 Tf 5.047 1.434 Td [(and)]TJ/F193 8.9265 Tf 16.321 0 Td [(B)]TJ/F158 6.6948 Tf 7.32 -1.434 Td [(f)]TJ/F108 5.8022 Tf 2.477 0 Td [(i)]TJ/F21 4.9734 Tf 1.659 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(i)]TJ/F108 4.9734 Tf 1.659 -1.005 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.005 Td [(g)]TJ/F21 8.2889 Tf 2.975 1.434 Td [(.)-250(Also,)-250(define)]TJ/F108 8.2889 Tf 50.372 0 Td [(W)]TJ/F158 6.6948 Tf 8.131 -1.434 Td [(;)]TJ/F158 9.5641 Tf 6.877 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(W)]TJ/F21 8.2889 Tf 9.246 0 Td [(,)]TJ/F156 9.5641 Tf 4.012 0 Td [(v)]TJ/F158 6.6948 Tf 4.658 -1.434 Td [(;)]TJ/F158 9.5641 Tf 6.877 1.434 Td [(D)]TJ/F156 9.5641 Tf 10.116 0 Td [(v)]TJ/F21 8.2889 Tf 5.041 0 Td [(,)]TJ/F193 8.9265 Tf 4.011 0 Td [(B)]TJ/F158 6.6948 Tf 7.32 -1.434 Td [(;)]TJ/F158 9.5641 Tf 6.877 1.434 Td [(D)]TJ/F193 8.9265 Tf 10.117 0 Td [(B)]TJ/F21 8.2889 Tf 7.891 0 Td [(.)]TJ ET 1 0 0 1 40.867 560.087 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -560.087 cm BT /F21 8.2889 Tf 52.822 549.028 Td [(By)-268(using)]TJ/F158 9.5641 Tf 34.677 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(U)]TJ/F21 8.2889 Tf 5.409 0 Td [(given)-268(by)]TJ ET 1 0 0 1 133.107 549.028 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -133.107 -549.028 cm BT /F21 8.2889 Tf 133.107 549.028 Td [(\0504.4\051)]TJ ET 1 0 0 1 150.779 549.028 cm 0 g 0 G 1 0 0 1 -150.779 -549.028 cm BT /F21 8.2889 Tf 150.779 549.028 Td [(,)-268(we)-269(construct)-268(the)-268(following)-268(mappings)-269(that)-268(commute)-268(with)-269(the)-268(action)-268(of)]TJ/F108 8.2889 Tf 275.527 0 Td [(x)]TJ/F156 6.6948 Tf 4.069 -1.434 Td [(\015)]TJ/F156 9.5641 Tf 4.93 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F156 9.5641 Tf 4.164 0 Td [(\015)]TJ/F158 9.5641 Tf 9.377 0 Td [(2)]TJ/F156 9.2055 Tf 9.061 0 Td [(\000)]TJ/F21 8.2889 Tf 7.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.164 0 Td [(n)]TJ/F158 9.5641 Tf 7.836 0 Td [(2)]TJ/F67 8.6077 Tf 9.062 0 Td [(Z)]TJ/F21 8.2889 Tf 5.738 0 Td [(:)]TJ ET 1 0 0 1 40.867 549.028 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -549.028 cm BT /F21 8.2889 Tf 40.867 538.567 Td [(for)]TJ/F108 8.2889 Tf 12.93 0 Td [(t)]TJ/F158 9.5641 Tf 6.3 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(0)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(m)]TJ/F158 9.5641 Tf 9.287 0 Td [(\000)]TJ/F21 8.2889 Tf 9.586 0 Td [(1,)-250(let)]TJ/F156 9.5641 Tf 20.813 0 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.434 Td [(t)]TJ/F21 8.2889 Tf 5.12 1.434 Td [(be)-250(a)]TJ/F108 8.2889 Tf 17.598 0 Td [(U)]TJ/F156 9.5641 Tf 6.502 0 Td [(.)]TJ/F158 9.5641 Tf 4.143 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 5.8022 Tf 4.335 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(-homogeneous)-250(mapping)]TJ ET 1 0 0 1 40.867 538.567 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -538.567 cm BT /F156 9.5641 Tf 64.777 519.304 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F158 9.5641 Tf 5.705 1.435 Td [(V)]TJ/F159 9.5641 Tf 10.847 9.086 Td [(X)]TJ/F108 5.8022 Tf -5.56 -19.69 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.004 Td [(t)]TJ/F156 6.6948 Tf 2.684 1.004 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.091 10.604 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F158 9.5641 Tf 5.838 2.439 Td [(!)]TJ/F159 9.5641 Tf 26.084 9.086 Td [(X)]TJ/F108 5.8022 Tf -12.916 -19.69 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.092 10.604 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ ET 1 0 0 1 40.867 519.304 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -519.304 cm BT /F21 8.2889 Tf 40.867 491.077 Td [(given)-250(component-wise)-250(by)]TJ ET 1 0 0 1 40.867 491.077 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -491.077 cm BT /F156 9.5641 Tf 64.777 471.872 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.434 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.435 Td [(.v)]TJ/F108 5.8022 Tf 8.225 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F156 9.5641 Tf 3.182 2.439 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F159 9.5641 Tf 16.586 9.085 Td [(X)]TJ/F158 5.7385 Tf -5.274 -19.032 Td [(f)]TJ/F108 4.9734 Tf 2.123 0 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(g2)]TJ/F108 4.9734 Tf 5.732 0 Td [(D)]TJ/F21 4.9734 Tf 3.452 -1.705 Td [(1)]TJ/F156 5.7385 Tf 3.378 1.705 Td [(.)]TJ/F108 4.9734 Tf 2.14 0 Td [(K)]TJ/F156 5.7385 Tf 3.693 0 Td [(/)]TJ/F108 4.9734 Tf -14.507 -6.456 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(6)1(2)]TJ/F108 4.9734 Tf 3.609 0 Td [(I)]TJ 1.497 -1.191 Td [(t)]TJ/F156 9.5641 Tf 11.314 17.594 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.949 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(I)]TJ 1.497 -1.19 Td [(t)]TJ/F156 6.6948 Tf 3.182 2.542 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F156 9.5641 Tf 2.996 -3.95 Td [(v)]TJ/F108 5.8022 Tf 4.657 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [([f)]TJ/F108 5.8022 Tf 6.949 0 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(g)]TJ/F156 9.5641 Tf 2.976 1.435 Td [(;)]TJ ET 1 0 0 1 40.867 471.872 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -471.872 cm BT /F21 8.2889 Tf 40.867 438.231 Td [(with)]TJ/F108 8.2889 Tf 19.793 0 Td [(p)]TJ/F108 5.8022 Tf 4.609 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F156 9.5641 Tf 3.182 2.439 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(i)]TJ/F156 9.5641 Tf 2.37 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.316 0 Td [(i)]TJ/F158 9.5641 Tf 5.028 0 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(j)]TJ/F108 5.8022 Tf 2.396 -1.435 Td [(p)]TJ/F108 4.9734 Tf 3.226 -1.352 Td [(I)]TJ 1.497 -1.191 Td [(t)]TJ/F156 6.6948 Tf 3.182 2.543 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(i)]TJ/F156 6.6948 Tf 1.659 0 Td [(/)]TJ/F21 8.2889 Tf 5.068 1.435 Td [(in)]TJ/F108 8.2889 Tf 9.582 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 9.5641 Tf 5.173 1.435 Td [([)-222(f)]TJ/F108 8.2889 Tf 12.053 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(g)-278(D)-278(f)]TJ/F108 8.2889 Tf 19.85 0 Td [(j)]TJ/F21 5.8022 Tf 2.396 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(j)]TJ/F108 5.8022 Tf 2.396 -1.435 Td [(t)]TJ/F158 9.5641 Tf 3.048 1.435 Td [(g)]TJ/F21 8.2889 Tf 5.611 0 Td [(for)-250(some)]TJ/F158 9.5641 Tf 35.443 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(j)]TJ/F21 5.8022 Tf 2.395 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(j)]TJ/F108 5.8022 Tf 2.396 -1.435 Td [(t)]TJ/F158 9.5641 Tf 3.048 1.435 Td [(g)-278(2)]TJ/F108 8.2889 Tf 14.868 0 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(.)]TJ ET 1 0 0 1 40.867 438.231 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -438.231 cm BT /F21 8.2889 Tf 52.822 426.044 Td [(Observe)-284(that)]TJ/F156 9.5641 Tf 51.243 0 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.435 Td [(t)]TJ/F21 8.2889 Tf 5.399 1.435 Td [(can)-284(be)-283(described)-284(also)-283(by)-284(providing)-284(the)-283(components)-284(of)-283(the)-284(image:)-284(for)]TJ/F108 8.2889 Tf 265.777 0 Td [(I)]TJ/F108 5.8022 Tf 2.495 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 7.228 1.435 Td [(D)-352(f)]TJ/F108 8.2889 Tf 14.37 0 Td [(i)]TJ/F21 5.8022 Tf 2.37 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(i)]TJ/F108 5.8022 Tf 2.37 -1.435 Td [(t)]TJ/F158 9.5641 Tf 3.049 1.435 Td [(g)-352(2)]TJ/F108 8.2889 Tf 16.296 0 Td [(D)]TJ/F108 5.8022 Tf 5.752 -1.435 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F21 8.2889 Tf 5.918 0 Td [(we)]TJ ET 1 0 0 1 40.867 426.044 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -426.044 cm BT /F21 8.2889 Tf 40.867 415.583 Td [(have)]TJ ET 1 0 0 1 40.867 415.583 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -415.583 cm BT /F156 9.5641 Tf 64.777 397.748 Td [(')]TJ/F108 5.8022 Tf 5.375 -1.434 Td [(t)]TJ/F156 9.5641 Tf 3.048 1.434 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 6.532 2.787 Td [(D)]TJ/F159 9.5641 Tf 12.537 9.086 Td [(X)]TJ/F21 5.8022 Tf -2.42 -19.543 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(s)]TJ/F158 6.6948 Tf 2.367 0 Td [(\024)]TJ/F108 5.8022 Tf 5.048 0 Td [(t)]TJ/F156 9.5641 Tf 2.55 10.457 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F108 5.8022 Tf 3.567 3.95 Td [(s)]TJ/F108 8.2889 Tf 2.866 -3.95 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.185 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(nf)]TJ/F108 5.8022 Tf 5.724 0 Td [(i)]TJ/F108 4.9734 Tf 1.66 -1.004 Td [(s)]TJ/F158 6.6948 Tf 2.527 1.004 Td [(g)]TJ/F156 9.5641 Tf 2.975 1.435 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.353 Td [(t)]TJ/F158 5.7385 Tf 2.186 0 Td [(C)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(;)]TJ ET 1 0 0 1 40.867 397.748 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -397.748 cm BT /F21 8.2889 Tf 40.867 369.788 Td [(where)]TJ/F108 8.2889 Tf 26.333 0 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.451 -1.434 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.005 Td [(t)]TJ/F21 8.2889 Tf 5.254 2.439 Td [(stands)-250(for)-250(the)-250(monomial)-250(part)-250(of)]TJ/F156 9.5641 Tf 122.75 0 Td [(w)]TJ/F158 9.5641 Tf 9.973 0 Td [(D)]TJ/F159 9.5641 Tf 10.116 7.173 Td [(P)]TJ/F108 5.8022 Tf 9.995 -9.899 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F158 6.6948 Tf 2.684 1.004 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F108 4.9734 Tf 4.027 -1.004 Td [(t)]TJ/F156 6.6948 Tf 2.684 1.004 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.589 2.726 Td [(a)]TJ/F156 9.5641 Tf 4.418 0 Td [(.w)-30(/)]TJ/F108 5.8022 Tf 14.452 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F156 9.5641 Tf 3.182 2.439 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F21 8.2889 Tf 5.255 2.439 Td [(in)]TJ/F108 8.2889 Tf 9.581 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(t)]TJ/F21 8.2889 Tf 3.182 2.439 Td [(.)]TJ ET 1 0 0 1 40.867 369.788 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -369.788 cm BT /F21 8.2889 Tf 52.822 358.729 Td [(We)-250(can)-250(now)-250(state)-250(the)-250(exactness)-250(result:)]TJ ET 1 0 0 1 40.867 358.729 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -22.07 cm 0 g 0 G 1 0 0 1 -36.882 -336.659 cm BT /F123 8.2889 Tf 40.867 336.659 Td [(Theorem)-250(5.1.)]TJ ET 1 0 0 1 92.813 336.659 cm 0 g 0 G 1 0 0 1 -92.813 -336.659 cm BT /F108 8.2889 Tf 96.798 336.659 Td [(For)-250(every)-250(K)]TJ/F158 9.5641 Tf 44.569 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.434 Td [(/)]TJ/F108 8.2889 Tf 5.64 0 Td [(such)-250(that)-335(k)]TJ/F21 5.8022 Tf 40.702 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.311 -1.434 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.434 Td [(D)]TJ/F108 8.2889 Tf 10.116 0 Td [(k)-250(the)-250(following)-250(sequence)-250(is)-250(exact:)]TJ ET 1 0 0 1 40.867 336.659 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -336.659 cm BT /F21 8.2889 Tf 64.777 313.385 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.018 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 8.578 8.87 Td [(T)]TJ/F156 6.6948 Tf 2.23 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 5.7385 Tf 2.229 2.43 Td [(\012)]TJ/F108 4.9734 Tf 4.591 0 Td [(k)]TJ/F158 9.5641 Tf -15.614 -8.473 Td [(\000)466(\000)222(\000)465(!)]TJ/F108 8.2889 Tf 24.516 0 Td [(W)]TJ/F156 6.6948 Tf 13.949 6.396 Td [(')]TJ/F21 4.9734 Tf 3.762 -1.353 Td [(0)]TJ/F158 9.5641 Tf -5.808 -5.043 Td [(\000)588(!)]TJ/F159 9.5641 Tf 21.258 9.086 Td [(X)]TJ/F108 5.8022 Tf -6.254 -19.69 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F21 4.9734 Tf 4.027 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.091 10.604 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F21 4.9734 Tf 1.747 -1.353 Td [(1)]TJ/F156 6.6948 Tf 8.578 9.14 Td [(')]TJ/F21 4.9734 Tf 3.762 -1.353 Td [(1)]TJ/F158 9.5641 Tf -5.808 -4.999 Td [(\000)588(!)-278(\001)-166(\001)-167(\001)]TJ/F156 6.6948 Tf 30.871 6.828 Td [(')]TJ/F108 4.9734 Tf 3.762 -1.353 Td [(m)]TJ/F158 5.7385 Tf 4.297 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf -14.581 -5.475 Td [(\000)225(\000)226(!)]TJ/F108 8.2889 Tf 23.777 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(m)]TJ/F158 9.5641 Tf 7.95 2.439 Td [(!)]TJ/F21 8.2889 Tf 13.167 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(:)]TJ ET 1 0 0 1 40.867 313.385 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -313.385 cm BT /F21 8.2889 Tf 52.822 277.254 Td [(Note)-250(that)-250(in)-250(the)-250(case)-250(of)]TJ/F108 8.2889 Tf 89.959 0 Td [(K)]TJ/F158 9.5641 Tf 8.811 0 Td [(D)]TJ/F156 9.5641 Tf 10.116 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F21 8.2889 Tf 4.76 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F21 8.2889 Tf 22.192 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(we)-250(have)]TJ/F108 8.2889 Tf 33.793 0 Td [(m)]TJ/F158 9.5641 Tf 9.818 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)-250(which)-250(yields)-250(the)-250(short)-250(exact)-250(sequence)]TJ ET 1 0 0 1 40.867 277.254 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -277.254 cm BT /F21 8.2889 Tf 64.777 253.783 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.434 Td [(k)]TJ/F156 6.6948 Tf 3.017 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F158 6.6948 Tf 8.56 7.478 Td [(T)]TJ/F156 6.6948 Tf 2.229 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 5.7385 Tf 2.23 2.43 Td [(\012)]TJ/F108 4.9734 Tf 4.59 0 Td [(k)]TJ/F158 9.5641 Tf -15.614 -8.474 Td [(\000)466(\000)222(\000)465(!)]TJ/F108 8.2889 Tf 24.517 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.434 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;:::;)]TJ/F21 5.8022 Tf 9.955 0 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F158 9.5641 Tf 6.172 1.434 Td [(!)]TJ/F21 8.2889 Tf 13.168 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(:)]TJ ET 1 0 0 1 40.867 253.783 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -31.679 cm 0 g 0 G 1 0 0 1 -36.882 -222.104 cm BT /F123 8.2889 Tf 40.867 222.104 Td [(Example)-250(5.2.)]TJ ET 1 0 0 1 91.437 222.104 cm 0 g 0 G 1 0 0 1 -91.437 -222.104 cm BT /F108 8.2889 Tf 95.422 222.104 Td [(For)]TJ/F156 9.5641 Tf 14.559 0 Td [(`)]TJ/F158 9.5641 Tf 7.133 0 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(2)]TJ/F108 8.2889 Tf 4.8 0 Td [(,)-250(k)]TJ/F158 9.5641 Tf 10.92 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(2)]TJ/F108 8.2889 Tf 6.871 0 Td [(the)-250(exact)-250(sequences)-250(are:)]TJ ET 1 0 0 1 40.867 222.104 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -222.104 cm BT /F21 8.2889 Tf 64.777 204.469 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 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2.216 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.434 Td [(!)]TJ/F21 8.2889 Tf 13.167 0 Td [(0)]TJ ET 1 0 0 1 40.867 191.019 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -191.019 cm BT /F21 8.2889 Tf 64.777 177.57 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(1)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(!)]TJ/F108 8.2889 Tf 13.168 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(!)]TJ/F21 8.2889 Tf 13.168 0 Td [(0)]TJ ET 1 0 0 1 40.867 177.57 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -177.57 cm BT /F21 8.2889 Tf 64.777 164.12 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F21 5.8022 Tf 8.132 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(2)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(0)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(!)]TJ/F108 8.2889 Tf 13.168 0 Td [(W)]TJ/F21 5.8022 Tf 8.131 -1.435 Td [(0)]TJ/F156 6.6948 Tf 3.359 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F156 6.6948 Tf 3.36 0 Td [(;)]TJ/F21 5.8022 Tf 2.216 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(!)]TJ/F21 8.2889 Tf 13.168 0 Td [(0)]TJ ET 1 0 0 1 40.867 164.12 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -164.12 cm BT /F21 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Tf 5.752 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(K)]TJ/F156 9.5641 Tf 6.154 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(\051)-432(will)-431(be)-432(used)-432(in)-431(the)-432(proof)-432(of)]TJ ET 1 0 0 1 40.867 111.215 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -10.461 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -40.867 -100.754 cm BT /F21 8.2889 Tf 40.867 100.754 Td [(Theorem)-250(5.1)]TJ ET 1 0 0 1 88.635 100.754 cm 0 g 0 G 1 0 0 1 -88.635 -100.754 cm BT /F21 8.2889 Tf 88.635 100.754 Td [(.)]TJ ET 1 0 0 1 40.867 100.754 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -22.069 cm 0 g 0 G 1 0 0 1 -36.882 -78.685 cm BT /F123 8.2889 Tf 40.867 78.685 Td [(Lemma)-250(5.3.)]TJ ET 1 0 0 1 86.522 78.685 cm 0 g 0 G 1 0 0 1 -86.522 -78.685 cm BT /F108 8.2889 Tf 90.507 78.685 Td [(For)-340(A)]TJ/F156 9.5641 Tf 19.59 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(B)]TJ/F158 9.5641 Tf 7.423 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(D)]TJ/F156 9.5641 Tf 5.753 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(K)]TJ/F156 9.5641 Tf 6.155 0 Td [(/)]TJ/F108 8.2889 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9.5641 Tf 5.043 0 Td [(\015)]TJ/F158 9.5641 Tf 11.629 0 Td [(2)]TJ/F156 9.2055 Tf 11.313 0 Td [(\000)]TJ/F21 8.2889 Tf 7.567 0 Td [(,)]TJ/F108 8.2889 Tf 5.043 0 Td [(n)]TJ/F158 9.5641 Tf 10.088 0 Td [(2)]TJ/F67 8.6077 Tf 11.314 0 Td [(Z)]TJ/F21 8.2889 Tf 5.738 0 Td [(.)-375(In)-374(particular,)-375(this)-374(holds)-374(for)]TJ ET 1 0 0 1 37.134 680.585 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -680.585 cm BT /F21 8.2889 Tf 37.134 670.125 Td [(coefficients)]TJ ET 1 0 0 1 37.134 670.125 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -670.125 cm BT /F158 9.5641 Tf 61.044 654.134 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(U)-278(D)]TJ/F21 8.2889 Tf 15.958 0 Td [(Res)]TJ/F108 8.2889 Tf 13.179 0 Td [(z)]TJ/F158 6.6948 Tf 4.265 3.95 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.359 0 Td [(\000h)]TJ/F156 6.6948 Tf 7.679 0 Td [(\025)]TJ/F108 4.9734 Tf 3.662 -1.352 Td [(i)]TJ/F156 6.6948 Tf 1.921 1.352 Td [(;!)]TJ/F108 4.9734 Tf 6.601 -1.353 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(i)]TJ/F108 8.2889 Tf 2.955 -3.95 Td [(c)]TJ/F108 5.8022 Tf 3.73 -1.435 Td [(i)]TJ/F193 8.9265 Tf 2.157 1.435 Td [(Y)]TJ/F156 9.5641 Tf 6.642 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\025)]TJ/F108 4.9734 Tf 3.662 -1.352 Td [(i)]TJ/F156 9.5641 Tf 2.419 -2.598 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/;)]TJ/F108 8.2889 Tf 16.616 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`;)]TJ ET 1 0 0 1 37.134 654.134 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -654.134 cm BT /F21 8.2889 Tf 37.134 638.144 Td [(which)-250(we)-250(also)-250(call)-250(intertwining)-250(operators.)]TJ ET 1 0 0 1 37.134 638.144 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -638.144 cm BT /F21 8.2889 Tf 49.089 627.687 Td [(We)-250(have)]TJ ET 1 0 0 1 37.134 627.687 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -627.687 cm BT /F108 8.2889 Tf 61.044 609.481 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F158 6.6948 Tf 8.269 6.044 Td [(T)]TJ/F21 5.8022 Tf 2.23 0 Td [(1)]TJ/F158 6.6948 Tf 3.359 0 Td [(U)]TJ/F158 9.5641 Tf -7.634 -6.044 Td [(\000)517(!)]TJ/F108 8.2889 Tf 15.682 0 Td [(L)]TJ/F156 9.5641 Tf 3.928 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F158 6.6948 Tf 8.27 6.044 Td [(T)]TJ/F21 5.8022 Tf 2.229 0 Td [(2)]TJ/F158 6.6948 Tf 3.359 0 Td [(U)]TJ/F158 9.5641 Tf -7.634 -6.044 Td [(\000)517(!)]TJ/F108 8.2889 Tf 15.682 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.434 Td [(2)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(/)]TJ/F158 6.6948 Tf 8.27 6.044 Td [(T)]TJ/F21 5.8022 Tf 2.229 0 Td [(3)]TJ/F158 6.6948 Tf 3.36 0 Td [(U)]TJ/F158 9.5641 Tf -7.635 -6.044 Td [(\000)517(!)-278(\001)-166(\001)-167(\001)]TJ/F158 6.6948 Tf 31.549 6.044 Td [(T)]TJ/F156 6.6948 Tf 2.229 0 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.36 0 Td [(U)]TJ/F158 9.5641 Tf -15.99 -6.044 Td [(\000)491(\000)222(\000)491(!)]TJ/F108 8.2889 Tf 24.037 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 6.6948 Tf 7.327 -1.434 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F158 6.6948 Tf 8.269 6.044 Td [(T)]TJ/F156 6.6948 Tf 2.23 0 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(U)]TJ/F158 9.5641 Tf -7.408 -6.044 Td [(\000)541(!)]TJ/F108 8.2889 Tf 15.455 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 6.6948 Tf 7.327 -1.434 Td [(`)]TJ/F156 9.5641 Tf 3.632 1.434 Td [(/;)]TJ ET 1 0 0 1 37.134 609.481 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -609.481 cm BT /F21 8.2889 Tf 37.134 593.491 Td [(and)-250(for)-250(suitably)-250(chosen)]TJ/F108 8.2889 Tf 90.539 0 Td [(c)]TJ/F108 5.8022 Tf 3.73 -1.435 Td [(i)]TJ/F21 8.2889 Tf 4.23 1.435 Td [(in)-250(the)-250(definition)-250(of)-250(intertwining)-250(operators)-250(also)]TJ ET 1 0 0 1 37.134 593.491 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -593.491 cm BT /F156 9.5641 Tf 61.044 573.612 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(0)]TJ/F158 6.6948 Tf 8.578 8.831 Td [(T)]TJ/F21 5.8022 Tf 2.23 0 Td [(1)]TJ/F158 6.6948 Tf 3.359 0 Td [(U)]TJ/F158 9.5641 Tf -7.634 -6.043 Td [(\000)517(!)]TJ/F156 9.5641 Tf 15.682 0 Td [(v)]TJ/F156 6.4438 Tf 4.657 -1.435 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(1)]TJ/F158 6.6948 Tf 8.579 8.831 Td [(T)]TJ/F21 5.8022 Tf 2.229 0 Td [(2)]TJ/F158 6.6948 Tf 3.359 0 Td [(U)]TJ/F158 9.5641 Tf -7.634 -6.043 Td [(\000)517(!)]TJ/F156 9.5641 Tf 15.682 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(2)]TJ/F158 6.6948 Tf 8.578 8.831 Td [(T)]TJ/F21 5.8022 Tf 2.229 0 Td [(3)]TJ/F158 6.6948 Tf 3.36 0 Td [(U)]TJ/F158 9.5641 Tf -7.635 -6.043 Td [(\000)517(!)-278(\001)-166(\001)-167(\001)]TJ/F158 6.6948 Tf 31.549 6.043 Td [(T)]TJ/F156 6.6948 Tf 2.229 0 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 6.6948 Tf 3.36 0 Td [(U)]TJ/F158 9.5641 Tf -15.99 -6.043 Td [(\000)491(\000)222(\000)491(!)]TJ/F156 9.5641 Tf 24.037 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F156 5.7385 Tf 5.129 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 8.578 8.87 Td [(T)]TJ/F156 6.6948 Tf 2.229 0 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(U)]TJ/F158 9.5641 Tf -7.408 -6.043 Td [(\000)541(!)]TJ/F156 9.5641 Tf 15.456 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F156 5.7385 Tf 5.129 -1.392 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(:)]TJ ET 1 0 0 1 37.134 573.612 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -573.612 cm BT /F21 8.2889 Tf 49.089 557.625 Td [(Next,)-250(define)-250(linear)-250(bijection)]TJ/F158 9.5641 Tf 107.805 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.455 0 Td [(U)]TJ/F21 8.2889 Tf 5.257 0 Td [(on)]TJ/F108 8.2889 Tf 11.795 0 Td [(M)]TJ/F156 9.5641 Tf 8.053 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F158 9.5641 Tf 5.692 0 Td [(\012)]TJ/F67 8.6077 Tf 9.777 0 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(P)]TJ/F158 9.5641 Tf 5.242 0 Td [(U)]TJ/F21 8.2889 Tf 5.257 0 Td [(by)]TJ ET 1 0 0 1 37.134 557.625 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -557.625 cm BT /F158 9.5641 Tf 61.044 541.635 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)-278(D)]TJ/F21 8.2889 Tf 15.958 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(!)]TJ/F156 5.7385 Tf 4.385 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 -2.557 Td [(\017)-55(.)]TJ/F158 9.5641 Tf 8.273 0 Td [(\001)]TJ/F156 9.5641 Tf 2.659 0 Td [(;)-167(!)]TJ/F156 6.6948 Tf 11.024 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/:)]TJ ET 1 0 0 1 37.134 541.635 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -541.635 cm BT /F21 8.2889 Tf 37.134 525.644 Td [(We)-250(call)]TJ/F158 9.5641 Tf 30.362 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F21 8.2889 Tf 5.257 0 Td [(a)-250(simple)-250(current)-250(operator)-250(\050cf.)-250([)]TJ ET 1 0 0 1 198.57 525.644 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -198.57 -525.644 cm BT /F21 8.2889 Tf 198.57 525.644 Td [(29)]TJ ET 1 0 0 1 208.168 525.644 cm 0 g 0 G 1 0 0 1 -208.168 -525.644 cm BT /F21 8.2889 Tf 208.168 525.644 Td [(]\051.)-250(It)-250(can)-250(be)-250(shown)-250(that)]TJ ET 1 0 0 1 37.134 525.644 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -525.644 cm BT /F108 8.2889 Tf 61.044 505.798 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F158 6.6948 Tf 8.269 6.043 Td [(T)]TJ/F156 6.6948 Tf 2.23 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 9.5641 Tf -8.794 -6.043 Td [(\000)396(!)]TJ/F108 8.2889 Tf 16.841 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/)]TJ/F158 6.6948 Tf 8.27 6.043 Td [(T)]TJ/F156 6.6948 Tf 2.229 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 9.5641 Tf -8.794 -6.043 Td [(\000)396(!)]TJ/F108 8.2889 Tf 16.842 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 6.6948 Tf 8.27 6.043 Td [(T)]TJ/F156 6.6948 Tf 2.229 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 9.5641 Tf -8.794 -6.043 Td [(\000)396(!)-278(\001)-167(\001)-166(\001)]TJ/F158 6.6948 Tf 32.708 6.043 Td [(T)]TJ/F156 6.6948 Tf 2.23 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 9.5641 Tf -8.794 -6.043 Td [(\000)396(!)]TJ/F108 8.2889 Tf 16.841 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F158 6.6948 Tf 8.269 6.043 Td [(T)]TJ/F156 6.6948 Tf 2.23 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 9.5641 Tf -8.794 -6.043 Td [(\000)396(!)]TJ/F108 8.2889 Tf 16.841 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/;)]TJ ET 1 0 0 1 37.134 505.798 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -505.798 cm BT /F21 8.2889 Tf 37.134 489.808 Td [(and)-250(from)-250(vertex)-250(operator)-250(formula)]TJ ET 1 0 0 1 167.783 489.808 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -167.783 -489.808 cm BT /F21 8.2889 Tf 167.783 489.808 Td [(\0504.1\051)]TJ ET 1 0 0 1 185.454 489.808 cm 0 g 0 G 1 0 0 1 -185.454 -489.808 cm BT /F21 8.2889 Tf 187.527 489.808 Td [(we)-250(have)]TJ ET 1 0 0 1 37.134 489.808 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -489.808 cm BT /F158 9.5641 Tf 61.044 473.817 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F156 9.5641 Tf 3.185 0 Td [(v)]TJ/F156 6.4438 Tf 4.657 -1.434 Td [(\003)]TJ/F21 4.9734 Tf 5.13 -1.353 Td [(0)]TJ/F158 9.5641 Tf 6.532 2.787 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.434 Td [(\003)]TJ/F156 5.7385 Tf 5.129 -1.393 Td [(`)]TJ/F156 9.5641 Tf 3.682 2.827 Td [(;)]TJ/F158 9.5641 Tf 13.048 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F156 9.5641 Tf 3.185 0 Td [(v)]TJ/F156 6.4438 Tf 4.657 -1.434 Td [(\003)]TJ/F108 4.9734 Tf 5.13 -1.352 Td [(i)]TJ/F158 9.5641 Tf 5.075 2.786 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(x)]TJ/F156 6.6948 Tf 4.069 -1.434 Td [(\015)]TJ/F108 4.9734 Tf 3.361 -1.352 Td [(i)]TJ/F156 9.5641 Tf 2.419 2.786 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/v)]TJ/F156 6.4438 Tf 8.225 -1.434 Td [(\003)]TJ/F108 4.9734 Tf 5.129 -1.353 Td [(i)]TJ/F158 5.7385 Tf 1.422 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.787 Td [(;)]TJ/F108 8.2889 Tf 13.049 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`:)]TJ/F21 8.2889 Tf 239.29 0 Td [(\0504.2\051)]TJ ET 1 0 0 1 37.134 473.817 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -473.817 cm BT /F21 8.2889 Tf 37.134 456.09 Td [(Also,)-256(from)]TJ ET 1 0 0 1 78.199 456.09 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -78.199 -456.09 cm BT /F21 8.2889 Tf 78.199 456.09 Td [(\0504.1\051)]TJ ET 1 0 0 1 95.871 456.09 cm 0 g 0 G 1 0 0 1 -95.871 -456.09 cm BT /F21 8.2889 Tf 97.99 456.09 Td [(it)-256(follows)]TJ/F108 8.2889 Tf 37.75 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\013)]TJ/F156 9.5641 Tf 4.589 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F158 9.5641 Tf 3.567 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)-290(D)-291(T)]TJ/F156 9.5641 Tf 19.384 0 Td [(!)]TJ/F158 9.5641 Tf 6.455 0 Td [(U)]TJ/F108 8.2889 Tf 3.185 0 Td [(z)]TJ/F158 6.6948 Tf 4.265 3.472 Td [(h)]TJ/F156 6.6948 Tf 2.457 0 Td [(!)]TJ/F156 5.7385 Tf 4.385 -1.393 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.393 Td [(;\013)]TJ/F158 6.6948 Tf 6.306 0 Td [(i)]TJ/F108 8.2889 Tf 2.955 -3.472 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\013)]TJ/F156 9.5641 Tf 4.589 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F21 8.2889 Tf 5.686 0 Td [(for)]TJ/F156 9.5641 Tf 12.978 0 Td [(\013)]TJ/F158 9.5641 Tf 8.62 0 Td [(2)]TJ/F108 8.2889 Tf 8.793 0 Td [(R)]TJ/F21 8.2889 Tf 6.985 0 Td [(or,)-256(written)-255(in)-256(components:)]TJ/F108 8.2889 Tf 104.718 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\013)]TJ/F156 9.5641 Tf 4.589 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F158 9.5641 Tf 3.568 0 Td [(T)]TJ/F156 9.5641 Tf 3.184 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)-290(D)-291(T)]TJ/F156 9.5641 Tf 19.384 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F108 8.2889 Tf 3.184 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\013)]TJ/F156 9.5641 Tf 4.589 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F158 9.5641 Tf 6.965 0 Td [(C)-227(h)]TJ/F156 9.5641 Tf 13.143 0 Td [(!)]TJ/F156 6.6948 Tf 6.265 -1.434 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.434 Td [(;)-167(\013)]TJ/F158 9.5641 Tf 10.603 0 Td [(i)]TJ/F156 9.5641 Tf 3.51 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)]TJ ET 1 0 0 1 37.134 456.09 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -456.09 cm BT /F156 9.5641 Tf 37.134 445.629 Td [(\013)]TJ/F158 9.5641 Tf 8.5 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(R)]TJ/F21 8.2889 Tf 4.865 0 Td [(,)]TJ/F108 8.2889 Tf 4.012 0 Td [(n)]TJ/F158 9.5641 Tf 7.448 0 Td [(2)]TJ/F67 8.6077 Tf 8.672 0 Td [(Z)]TJ/F21 8.2889 Tf 5.739 0 Td [(,)-250(which)-250(specially)-250(for)]TJ/F156 9.5641 Tf 77.973 0 Td [(\015)]TJ/F158 9.5641 Tf 8.988 0 Td [(2)]TJ/F156 9.2055 Tf 8.672 0 Td [(\000)]TJ/F21 8.2889 Tf 9.639 0 Td [(gives)]TJ ET 1 0 0 1 37.134 445.629 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -445.629 cm BT /F108 8.2889 Tf 61.044 428.57 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\015)]TJ/F156 9.5641 Tf 4.93 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(n)]TJ/F156 9.5641 Tf 4.79 0 Td [(/)]TJ/F158 9.5641 Tf 3.568 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.455 0 Td [(U)-278(D)-278(T)]TJ/F156 9.5641 Tf 19.144 0 Td [(!)]TJ/F158 9.5641 Tf 6.455 0 Td [(U)]TJ/F108 8.2889 Tf 3.185 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\015)]TJ/F156 9.5641 Tf 4.93 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F158 9.5641 Tf 6.917 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/:)]TJ/F21 8.2889 Tf 331.452 0 Td [(\0504.3\051)]TJ ET 1 0 0 1 37.134 428.57 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -428.57 cm BT /F21 8.2889 Tf 37.134 412.58 Td [(In)-250(general,)-250(for)-250(every)]TJ/F108 8.2889 Tf 78.238 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F156 6.4438 Tf 18.487 -1.435 Td [(\003)]TJ/F21 8.2889 Tf 7.7 1.435 Td [(we)-250(have)]TJ ET 1 0 0 1 37.134 412.58 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -412.58 cm BT /F158 9.5641 Tf 61.044 396.589 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F108 8.2889 Tf 3.185 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F158 9.5641 Tf 16.486 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)]TJ/F158 6.6948 Tf 10.263 3.95 Td [(\000)]TJ/F156 9.5641 Tf 5.72 -3.95 Td [(/)]TJ/F158 9.5641 Tf 3.567 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F156 9.5641 Tf 3.185 0 Td [(;)]TJ ET 1 0 0 1 37.134 396.589 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -396.589 cm BT /F21 8.2889 Tf 37.134 380.599 Td [(where)]TJ/F108 8.2889 Tf 26.334 0 Td [(x)]TJ/F156 9.5641 Tf 4.069 0 Td [(.\031)]TJ/F158 6.6948 Tf 10.263 3.471 Td [(\000)]TJ/F156 9.5641 Tf 5.72 -3.471 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(denotes)-250(monomial)-250(obtained)-250(from)]TJ/F108 8.2889 Tf 129.729 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 15.902 0 Td [(by)-250(lowering)-250(the)-250(degree)-250(of)-250(every)-250(constituting)-250(factor)-250(in)]TJ/F108 8.2889 Tf 204.9 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 15.902 0 Td [(by)-250(one.)]TJ ET 1 0 0 1 37.134 380.599 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -380.599 cm BT /F108 8.2889 Tf 37.134 357.468 Td [(4.2.)-500(Higher)-250(level)-250(standard)-250(modules)]TJ ET 1 0 0 1 37.134 357.468 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -357.468 cm BT /F21 8.2889 Tf 49.089 336.567 Td [(Because)-273(of)-274(complete)-273(reducibility)-274(of)-273(tensor)-273(product)-274(of)-273(standard)-274(modules)-273(we)-274(can)-273(embed)-273(level)]TJ/F108 8.2889 Tf 354.257 0 Td [(k)]TJ/F21 8.2889 Tf 6.577 0 Td [(standard)]TJ/F158 9.5641 Tf 36.502 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(-module)]TJ/F108 8.2889 Tf 34.808 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ ET 1 0 0 1 37.134 336.567 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -336.567 cm BT /F21 8.2889 Tf 37.134 326.106 Td [(with)]TJ/F156 9.2055 Tf 19.794 0 Td [(\003)]TJ/F158 9.5641 Tf 9.984 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.857 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.632 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.327 -1.435 Td [(`)]TJ/F21 8.2889 Tf 5.704 1.435 Td [(in)-250(the)-250(appropriate)]TJ/F108 8.2889 Tf 71.093 0 Td [(k)]TJ/F21 8.2889 Tf 4.31 0 Td [(-fold)-250(tensor)-250(product)-250(of)-250(fundamental)-250(modules)]TJ ET 1 0 0 1 37.134 326.106 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -326.106 cm BT /F108 8.2889 Tf 61.044 310.115 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 9.5641 Tf 7.327 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(\032)]TJ/F108 8.2889 Tf 10.117 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.434 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.434 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 9.5641 Tf 5.808 -2.558 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F108 8.2889 Tf 32.843 0 Td [(L)]TJ/F156 9.5641 Tf 3.928 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F158 9.5641 Tf 6.002 -2.597 Td [(\012)]TJ/F108 8.2889 Tf 9.776 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 9.5641 Tf 3.876 -2.597 Td [(;)]TJ ET 1 0 0 1 37.134 310.115 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -310.115 cm BT /F21 8.2889 Tf 37.134 294.125 Td [(and)-250(we)-250(can)-250(take)-250(the)-250(corresponding)-250(highest)-250(weight)-250(vector)]TJ/F156 9.5641 Tf 220.392 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F21 8.2889 Tf 7.699 1.435 Td [(to)-250(be)]TJ ET 1 0 0 1 37.134 294.125 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -294.125 cm BT /F156 9.5641 Tf 61.044 276.412 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F158 9.5641 Tf 8.284 1.435 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(v)]TJ/F158 6.6948 Tf 5.04 4.986 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.4438 Tf -8.756 -6.087 Td [(\003)]TJ/F156 5.7385 Tf 5.13 -1.393 Td [(`)]TJ/F158 9.5641 Tf 9.433 3.886 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F156 9.5641 Tf 32.843 0 Td [(v)]TJ/F158 6.6948 Tf 5.04 4.872 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.4438 Tf -8.755 -6.012 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(1)]TJ/F158 9.5641 Tf 9.627 3.846 Td [(\012)]TJ/F156 9.5641 Tf 9.777 0 Td [(v)]TJ/F158 6.6948 Tf 5.04 4.917 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.4438 Tf -8.755 -6.057 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(0)]TJ/F156 9.5641 Tf 7.502 3.846 Td [(:)]TJ ET 1 0 0 1 37.134 276.412 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -276.412 cm BT /F21 8.2889 Tf 49.089 260.425 Td [(Denote)-250(again)]TJ/F158 9.5641 Tf 52.211 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(U)-278(V)56(D)]TJ/F21 8.2889 Tf 18.057 0 Td [(1)]TJ/F158 9.5641 Tf 6.924 0 Td [(\012)-222(\001)-167(\001)-167(\001)]TJ/F21 8.2889 Tf 22.536 0 Td [(1)]TJ/F158 9.5641 Tf 6.924 0 Td [(\012)-222(T)]TJ/F108 8.2889 Tf 12.962 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(U)-222(\012)]TJ/F21 8.2889 Tf 15.087 0 Td [(1)]TJ/F158 9.5641 Tf 6.393 0 Td [(\001)-167(\001)-166(\001)-223(\012)]TJ/F21 8.2889 Tf 23.067 0 Td [(1,)]TJ/F108 8.2889 Tf 8.811 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F21 8.2889 Tf 26.668 0 Td [(,)-250(i.e.)-250(the)]TJ/F108 8.2889 Tf 31.373 0 Td [(k)]TJ/F21 8.2889 Tf 4.31 0 Td [(-fold)-250(tensor)-250(product)-250(of)]TJ/F158 9.5641 Tf 88.326 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(i)]TJ/F158 9.5641 Tf 2.37 0 Td [(U)]TJ/F21 8.2889 Tf 5.257 0 Td [(with)-250(identity)-250(maps:)]TJ ET 1 0 0 1 37.134 260.425 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -260.425 cm BT /F158 9.5641 Tf 61.044 244.435 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(i)]TJ/F158 9.5641 Tf 2.371 0 Td [(U)-278(V)]TJ/F108 8.2889 Tf 11.128 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.568 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.355 0 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.393 Td [(`)]TJ/F158 9.5641 Tf 5.807 -2.557 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F108 8.2889 Tf 32.843 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F108 5.8022 Tf 7.328 -1.435 Td [(i)]TJ/F156 9.5641 Tf 2.157 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.568 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.355 0 Td [(k)]TJ/F108 4.9734 Tf 3.018 -1.353 Td [(i)]TJ/F158 9.5641 Tf 4.543 -2.597 Td [(\012)]TJ/F108 8.2889 Tf 9.777 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F108 5.8022 Tf 7.328 -1.435 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.353 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 6.001 -2.597 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F108 8.2889 Tf 32.843 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.568 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ ET 1 0 0 1 37.134 244.435 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -244.435 cm BT /F158 9.5641 Tf 69.333 228.064 Td [(!)]TJ/F108 8.2889 Tf 13.168 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 9.5641 Tf 5.807 -2.557 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F108 8.2889 Tf 32.843 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F108 5.8022 Tf 7.327 -1.435 Td [(i)]TJ/F156 9.5641 Tf 2.158 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.352 Td [(i)]TJ/F158 6.6948 Tf 1.921 1.352 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 -3.95 Td [(\012)]TJ/F108 8.2889 Tf 9.776 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F108 5.8022 Tf 7.328 -1.435 Td [(i)]TJ/F158 6.6948 Tf 1.659 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F108 4.9734 Tf 3.017 -1.353 Td [(i)]TJ/F158 5.7385 Tf 1.423 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 -3.95 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F108 8.2889 Tf 32.843 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F21 8.2889 Tf 160.895 -2.597 Td [(\0504.4\051)]TJ ET 1 0 0 1 37.134 228.064 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -228.064 cm BT /F21 8.2889 Tf 37.134 212.074 Td [(for)]TJ/F108 8.2889 Tf 12.964 0 Td [(k)]TJ/F108 5.8022 Tf 4.31 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.899 1.435 Td [(\025)]TJ/F21 8.2889 Tf 9.953 0 Td [(1.)-254(These)-254(are)-254(again)-254(linear)-254(maps)-254(between)-254(corresponding)-254(standard)-254(level)]TJ/F108 8.2889 Tf 265.259 0 Td [(k)]TJ/F158 9.5641 Tf 6.813 0.174 Td [(Q)]TJ/F108 8.2889 Tf -0.909 -0.174 Td [(g)]TJ/F21 8.2889 Tf 5.002 0 Td [(-modules)-254(that)-254(map)-254(highest)-254(weight)-254(vector)]TJ ET 1 0 0 1 37.134 212.074 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -212.074 cm BT /F21 8.2889 Tf 37.134 201.613 Td [(into)-250(highest)-250(weight)-250(vector.)-250(Also,)-250(these)-250(maps)-250(commute)-250(with)-250(action)-250(of)]TJ/F108 8.2889 Tf 263.469 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 9.5641 Tf 4.93 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)]TJ/F156 9.5641 Tf 4.011 0 Td [(\015)]TJ/F158 9.5641 Tf 8.989 0 Td [(2)]TJ/F156 9.2055 Tf 8.672 0 Td [(\000)]TJ/F21 8.2889 Tf 7.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.012 0 Td [(n)]TJ/F158 9.5641 Tf 7.447 0 Td [(2)]TJ/F67 8.6077 Tf 8.673 0 Td [(Z)]TJ/F21 8.2889 Tf 5.738 0 Td [(.)]TJ ET 1 0 0 1 37.134 201.613 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -201.613 cm BT /F21 8.2889 Tf 49.089 189.883 Td [(On)]TJ/F108 8.2889 Tf 13.431 0 Td [(k)]TJ/F21 8.2889 Tf 4.31 0 Td [(-fold)-298(tensor)-299(products)-298(of)-298(standard)]TJ/F158 9.5641 Tf 130.336 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(-modules)-298(we)-299(use)-298(also)]TJ/F158 9.5641 Tf 85.818 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F158 6.6948 Tf 3.184 3.472 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 8.2889 Tf 5.989 -3.472 Td [(\050we)-298(will)-299(denote)-298(it)-298(again)]TJ/F158 9.5641 Tf 94.109 0 Td [(T)]TJ/F156 9.5641 Tf 3.185 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F21 8.2889 Tf 3.185 0 Td [(\051,)-298(a)-299(linear)-298(bijection)-298(for)]TJ ET 1 0 0 1 37.134 189.883 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -189.883 cm BT /F21 8.2889 Tf 37.134 179.423 Td [(which)-250(the)-250(commutation)-250(formula)-250(analogous)-250(to)]TJ ET 1 0 0 1 214.117 179.423 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -214.117 -179.423 cm BT /F21 8.2889 Tf 214.117 179.423 Td [(\0504.3\051)]TJ ET 1 0 0 1 231.789 179.423 cm 0 g 0 G 1 0 0 1 -231.789 -179.423 cm BT /F21 8.2889 Tf 233.861 179.423 Td [(holds.)]TJ ET 1 0 0 1 37.134 179.423 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -179.423 cm BT /F123 8.2889 Tf 37.134 156.308 Td [(5.)-500(Exact)-250(sequences)-250(of)-250(Feigin\025Stoyanovsky's)-250(type)-250(subspaces)]TJ ET 1 0 0 1 37.134 156.308 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -156.308 cm BT /F108 8.2889 Tf 37.134 133.176 Td [(5.1.)-500(Exactness)-250(and)-250(supplementary)-250(results)]TJ ET 1 0 0 1 37.134 133.176 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -133.176 cm BT /F21 8.2889 Tf 49.089 112.275 Td [(For)-250(fixed)-250(level)]TJ/F108 8.2889 Tf 55.842 0 Td [(k)]TJ/F21 8.2889 Tf 6.382 0 Td [(and)-250(arbitrary)-250(nonnegative)-250(integers)]TJ/F108 8.2889 Tf 134.495 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F21 8.2889 Tf 5.703 1.435 Td [(such)-250(that)]TJ/F108 8.2889 Tf 37.317 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.435 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 8.2889 Tf 6.382 0 Td [(denote)]TJ ET 1 0 0 1 37.134 112.275 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -112.275 cm BT /F108 8.2889 Tf 61.044 96.284 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 9.5641 Tf 6.339 2.827 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.982 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.586 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(1)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.311 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.631 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.327 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.632 1.435 Td [(/;)]TJ ET 1 0 0 1 37.134 96.284 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -96.284 cm BT /F156 9.5641 Tf 61.044 79.959 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 9.5641 Tf 6.338 2.828 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(v)]TJ/F108 5.8022 Tf 4.658 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.4438 Tf 3.378 1.353 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(0)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(C)]TJ/F108 5.8022 Tf 5.222 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.4438 Tf 3.377 1.353 Td [(\003)]TJ/F21 4.9734 Tf 5.13 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.377 1.353 Td [(C\001\001\001C)]TJ/F108 5.8022 Tf 16.028 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F156 6.4438 Tf 3.184 1.393 Td [(\003)]TJ/F156 5.7385 Tf 5.129 -1.393 Td [(`)]TJ/F158 9.5641 Tf 6.338 2.828 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(v)]TJ/F158 6.6948 Tf 5.04 4.986 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.4438 Tf -8.755 -6.087 Td [(\003)]TJ/F156 5.7385 Tf 5.129 -1.393 Td [(`)]TJ/F158 9.5641 Tf 9.433 3.886 Td [(\012)-222(\001)-167(\001)-167(\001)-222(\012)]TJ/F156 9.5641 Tf 32.843 0 Td [(v)]TJ/F158 6.6948 Tf 5.041 4.872 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.4438 Tf -8.756 -6.012 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(1)]TJ/F158 9.5641 Tf 9.628 3.846 Td [(\012)]TJ/F156 9.5641 Tf 9.777 0 Td [(v)]TJ/F158 6.6948 Tf 5.04 4.917 Td [(\012)]TJ/F108 5.8022 Tf 5.356 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.4438 Tf -8.756 -6.057 Td [(\003)]TJ/F21 4.9734 Tf 5.129 -1.353 Td [(0)]TJ/F156 9.5641 Tf 7.503 3.846 Td [(:)]TJ ET 1 0 0 1 37.134 79.959 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -79.959 cm BT /F21 8.2889 Tf 37.134 62.902 Td [(Also,)-400(let)]TJ/F193 8.9265 Tf 34.838 0 Td [(B)]TJ/F108 5.8022 Tf 7.319 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F21 8.2889 Tf 6.993 2.828 Td [(be)-399(the)-400(set)-399(of)]TJ/F156 9.5641 Tf 53.485 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(;)-167(`)]TJ/F158 9.5641 Tf 12.632 0 Td [(C)]TJ/F21 8.2889 Tf 10.855 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(-admissible)-399(monomials)]TJ/F108 8.2889 Tf 93.687 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 17.14 0 Td [(for)]TJ/F108 8.2889 Tf 14.169 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F158 9.5641 Tf 7.253 1.435 Td [(C)]TJ/F108 8.2889 Tf 10.855 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(1)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(1)]TJ/F158 9.5641 Tf 7.254 1.435 Td [(C)-355(\001)-167(\001)-166(\001)-355(C)]TJ/F108 8.2889 Tf 36.271 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.631 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F21 8.2889 Tf 3.631 1.435 Td [(.)-400(Then)-400(vectors)]TJ ET 1 0 0 1 37.134 62.902 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -62.902 cm BT /F108 8.2889 Tf 37.134 52.441 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F108 5.8022 Tf 18.487 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F21 8.2889 Tf 5.754 2.827 Td [(belong)-250(to)-250(basis)-250(of)]TJ/F108 8.2889 Tf 68.366 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.956 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F21 8.2889 Tf 3.682 2.827 Td [(.)]TJ ET 1 0 0 1 37.134 52.441 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -11.955 cm 0 g 0 G 1 0 0 1 466.251 0 cm 0 g 0 G endstream endobj 167 0 obj << /Type /XObject /Subtype /Form /FormType 1 /PTEX.FileName (./main.pdf) /PTEX.PageNumber 4 /PTEX.InfoDict 124 0 R /Matrix [ 1 0 0 1 0 0 ] /BBox [ 0 0 544.252 742.677 ] /Resources << /Font << /F123 121 0 R /F156 120 0 R /F80 119 0 R /F193 43 0 R /F67 118 0 R /F21 117 0 R /F108 115 0 R /F158 113 0 R /F159 112 0 R >> /ProcSet [ /PDF /Text ] >> /Length 32437 >> stream 1 0 0 1 40.867 702.503 cm 0 g 0 G 1 0 0 1 -40.867 -702.503 cm BT /F21 6.6311 Tf 40.867 702.503 Td [(916)]TJ/F108 6.6311 Tf 131.912 0 Td [(M.)-250(Jerkovi\242)-250(/)-250(Journal)-250(of)-250(Pure)-250(and)-250(Applied)-250(Algebra)-250(213)-250(\0502009\051)-250(913\025926)]TJ ET 1 0 0 1 507.118 702.503 cm 0 g 0 G 1 0 0 1 -507.118 -702.503 cm BT /F108 8.2889 Tf 40.867 680.585 Td [(3.2.)-500(Combinatorial)-250(basis)-250(for)-250(W)]TJ/F156 9.5641 Tf 112.964 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ ET 1 0 0 1 40.867 680.585 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -680.585 cm BT /F21 8.2889 Tf 52.822 659.723 Td [(For)-254(given)]TJ/F156 9.2055 Tf 37.383 0 Td [(\003)]TJ/F158 9.5641 Tf 10.069 0 Td [(D)]TJ/F108 8.2889 Tf 10.202 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.434 Td [(0)]TJ/F156 9.2055 Tf 3.857 1.434 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.434 Td [(0)]TJ/F158 9.5641 Tf 6.017 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.619 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.434 Td [(1)]TJ/F156 9.2055 Tf 3.857 1.434 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.434 Td [(1)]TJ/F158 9.5641 Tf 6.017 1.434 Td [(C)-226(\001)-166(\001)-167(\001)-226(C)]TJ/F108 8.2889 Tf 32.563 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F156 9.2055 Tf 3.631 1.434 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.434 Td [(`)]TJ/F21 8.2889 Tf 5.737 1.434 Td [(a)-254(monomial)]TJ/F108 8.2889 Tf 47.553 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 15.935 0 Td [(given)-254(by)]TJ ET 1 0 0 1 318.655 659.723 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -318.655 -659.723 cm BT /F21 8.2889 Tf 318.655 659.723 Td [(\0503.1\051)]TJ ET 1 0 0 1 336.327 659.723 cm 0 g 0 G 1 0 0 1 -336.327 -659.723 cm BT /F21 8.2889 Tf 336.327 659.723 Td [(,)-254(or)-254(a)-254(monomial)-254(vector)]TJ/F108 8.2889 Tf 87.921 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/v)]TJ/F156 6.4438 Tf 18.487 -1.434 Td [(\003)]TJ/F21 8.2889 Tf 7.733 1.434 Td [(for)-254(such)]TJ/F108 8.2889 Tf 32.741 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.\031)-100(/)]TJ/F21 8.2889 Tf 13.83 0 Td [(,)]TJ ET 1 0 0 1 40.867 659.723 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -659.723 cm BT /F21 8.2889 Tf 40.867 649.262 Td [(is)-250(called)]TJ/F156 9.5641 Tf 32.84 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F156 9.5641 Tf 4.311 0 Td [(;)-167(`)]TJ/F158 9.5641 Tf 11.361 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(-admissible)-250(for)]TJ/F156 9.2055 Tf 59.323 0 Td [(\003)]TJ/F21 8.2889 Tf 9.4 0 Td [(\050or)]TJ/F108 8.2889 Tf 13.237 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(\051)-250(if)-250(the)-250(following)-250(inequalities)-250(are)-250(met:)]TJ ET 1 0 0 1 40.867 649.262 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -649.262 cm BT /F108 8.2889 Tf 64.777 631.766 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.434 Td [(0)]TJ/F158 9.5641 Tf 6.514 1.434 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F108 8.2889 Tf -25.11 -12.015 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.586 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(1)]TJ/F156 9.5641 Tf -64.975 -12.015 Td [(:)-167(:)-166(:)]TJ/F108 8.2889 Tf 0 -13.449 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(a)]TJ/F21 5.8022 Tf 4.418 -1.435 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(a)]TJ/F156 6.6948 Tf 4.418 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.515 1.435 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.134 0 Td [(\000)]TJ/F21 5.8022 Tf 5.221 0 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F21 8.2889 Tf 273.39 19.875 Td [(\0503.2\051)]TJ ET 1 0 0 1 40.867 611.293 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -611.293 cm BT /F21 8.2889 Tf 40.867 575.486 Td [(and)]TJ ET 1 0 0 1 40.867 575.486 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -575.486 cm BT /F108 8.2889 Tf 64.777 559.588 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.434 Td [(i)]TJ/F158 9.5641 Tf 4.283 1.434 Td [(C)-222(\001)-167(\001)-167(\001)]TJ/F108 8.2889 Tf 22.343 0 Td [(a)]TJ/F108 5.8022 Tf 4.418 -1.434 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(C)]TJ/F156 6.6948 Tf 5.222 0 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.434 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(;)]TJ/F108 8.2889 Tf 13.049 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(2)]TJ/F67 8.6077 Tf 8.672 0 Td [(Z)]TJ/F158 6.6948 Tf 5.739 -1.434 Td [(C)]TJ/F156 9.5641 Tf 5.72 1.434 Td [(:)]TJ/F21 8.2889 Tf 323.652 0 Td [(\0503.3\051)]TJ ET 1 0 0 1 40.867 559.588 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -559.588 cm BT /F21 8.2889 Tf 52.822 543.703 Td [(We)-250(say)-250(that)]TJ ET 1 0 0 1 100.064 543.703 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -100.064 -543.703 cm BT /F21 8.2889 Tf 100.064 543.703 Td [(\0503.2\051)]TJ ET 1 0 0 1 117.736 543.703 cm 0 g 0 G 1 0 0 1 -117.736 -543.703 cm BT /F21 8.2889 Tf 119.807 543.703 Td [(are)-250(the)-250(initial)-250(conditions)-249(for)-250(given)]TJ/F156 9.2055 Tf 131.122 0 Td [(\003)]TJ/F21 8.2889 Tf 9.399 0 Td [(\050or)]TJ/F108 8.2889 Tf 13.236 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(\051,)-250(and)-250(that)]TJ ET 1 0 0 1 338.232 543.703 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -338.232 -543.703 cm BT /F21 8.2889 Tf 338.232 543.703 Td [(\0503.3\051)]TJ ET 1 0 0 1 355.904 543.703 cm 0 g 0 G 1 0 0 1 -355.904 -543.703 cm BT /F21 8.2889 Tf 357.975 543.703 Td [(are)-250(the)-250(difference)-250(con)1(ditions)-250(\050note)-250(that)]TJ ET 1 0 0 1 40.867 543.703 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -543.703 cm BT /F21 8.2889 Tf 40.867 533.242 Td [(the)-250(difference)-250(conditions)]TJ ET 1 0 0 1 138.078 533.242 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -138.078 -533.242 cm BT /F21 8.2889 Tf 138.078 533.242 Td [(\0503.3\051)]TJ ET 1 0 0 1 155.75 533.242 cm 0 g 0 G 1 0 0 1 -155.75 -533.242 cm BT /F21 8.2889 Tf 157.822 533.242 Td [(do)-250(not)-250(depend)-250(on)-250(the)-250(choice)-250(of)]TJ/F156 9.2055 Tf 119.683 0 Td [(\003)]TJ/F21 8.2889 Tf 7.327 0 Td [(\051.)]TJ ET 1 0 0 1 40.867 533.242 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -533.242 cm BT /F21 8.2889 Tf 52.822 522.793 Td [(In)-174([)]TJ ET 1 0 0 1 64.851 522.793 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -64.851 -522.793 cm BT /F21 8.2889 Tf 64.851 522.793 Td [(19)]TJ ET 1 0 0 1 74.449 522.793 cm 0 g 0 G 1 0 0 1 -74.449 -522.793 cm BT /F21 8.2889 Tf 74.449 522.793 Td [(])-174(Primc)-174(demonstrated,)-175(by)-174(using)-174(certain)-174(coefficients)-174(of)-175(intertwining)-174(operators)-174(between)-174(fundamental)-174(modules)-175(as)-174(well)]TJ ET 1 0 0 1 40.867 522.793 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -522.793 cm BT /F21 8.2889 Tf 40.867 512.333 Td [(as)-283(a)-282(simple)-283(current)-283(operator,)-283(that)-283(the)-282(spanning)-283(set)-283(for)]TJ/F108 8.2889 Tf 209.423 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 9.5641 Tf 7.327 0 Td [(/)]TJ/F21 8.2889 Tf 5.911 0 Td [(given)-283(in)-283(3.1)-282(can)-283(be)-283(reduced)-283(to)-282(basis)-283(consisting)-283(of)]TJ/F156 9.5641 Tf 189.622 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(;)-167(`)]TJ/F158 9.5641 Tf 11.64 0 Td [(C)]TJ/F21 8.2889 Tf 9.864 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(-)]TJ ET 1 0 0 1 40.867 512.333 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -512.333 cm BT /F21 8.2889 Tf 40.867 501.872 Td [(admissible)-250(vectors:)]TJ ET 1 0 0 1 40.867 501.872 cm 0 g 0 G 0 g 0 G 1 0 0 1 -3.985 -18.888 cm 0 g 0 G 1 0 0 1 -36.882 -482.984 cm BT /F123 8.2889 Tf 40.867 482.984 Td [(Theorem)-250(3.1.)]TJ ET 1 0 0 1 92.813 482.984 cm 0 g 0 G 1 0 0 1 -92.813 -482.984 cm BT /F108 8.2889 Tf 96.798 482.984 Td [(The)-250(set)-250(of)]TJ/F156 9.5641 Tf 38.041 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(;)-167(`)]TJ/F158 9.5641 Tf 11.361 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(-admissible)-250(monomial)-250(vectors)-250(x)]TJ/F156 9.5641 Tf 115.14 0 Td [(.\031)-100(/v)]TJ/F156 6.4438 Tf 18.488 -1.435 Td [(\003)]TJ/F108 8.2889 Tf 7.699 1.435 Td [(is)-250(a)-250(basis)-250(of)-453(W)]TJ/F156 9.5641 Tf 54.342 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(.)]TJ ET 1 0 0 1 40.867 482.984 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -482.984 cm BT /F21 8.2889 Tf 52.822 466.652 Td [(Note)-289(that)-290(the)-289(formulation)-290(of)]TJ ET 1 0 0 1 164.108 466.652 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -164.108 -466.652 cm BT /F21 8.2889 Tf 164.108 466.652 Td [(Theorem)-289(3.1)]TJ ET 1 0 0 1 212.203 466.652 cm 0 g 0 G 1 0 0 1 -212.203 -466.652 cm BT /F21 8.2889 Tf 214.602 466.652 Td [(in)-289(terms)-290(of)]TJ/F156 9.5641 Tf 44.397 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(;)-167(`)]TJ/F158 9.5641 Tf 11.696 0 Td [(C)]TJ/F21 8.2889 Tf 9.921 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(-admissible)-289(monomials)-290(was)-289(first)-290(given)-289(in)-289([)]TJ ET 1 0 0 1 460.219 466.652 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -460.219 -466.652 cm BT /F21 8.2889 Tf 460.219 466.652 Td [(16)]TJ ET 1 0 0 1 469.818 466.652 cm 0 g 0 G 1 0 0 1 -469.818 -466.652 cm BT /F21 8.2889 Tf 469.818 466.652 Td [(].)-290(We)-289(will)]TJ ET 1 0 0 1 40.867 466.652 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -466.652 cm BT /F21 8.2889 Tf 40.867 456.192 Td [(not)-250(give)-250(details)-250(of)-250(the)-250(proof)-250(here,)-250(although)-250(some)-250(of)-250(the)-250(content)-250(needed)-250(for)-250(it)-250(is)-250(to)-250(be)-250(presented)-250(in)-250(the)-250(the)-250(next)-250(section.)]TJ ET 1 0 0 1 40.867 456.192 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -456.192 cm BT /F123 8.2889 Tf 40.867 433.178 Td [(4.)-500(Intertwining)-250(operators)-250(and)-250(simple)-250(current)-250(operator)]TJ ET 1 0 0 1 40.867 433.178 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -433.178 cm BT /F108 8.2889 Tf 40.867 410.116 Td [(4.1.)-500(Fundamental)-250(modules)]TJ ET 1 0 0 1 40.867 410.116 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -410.116 cm BT /F21 8.2889 Tf 52.822 389.254 Td [(In)-250(this)-250(section)-250(we)-250(state)-250(some)-250(facts)-250(needed)-250(in)-250(the)-250(following)-250(sections.)-250(For)-250(precise)-250(definitions)-250(and)-250(proofs)-250(see)-250([)]TJ ET 1 0 0 1 459.232 389.254 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -459.232 -389.254 cm BT /F21 8.2889 Tf 459.232 389.254 Td [(21\02526)]TJ ET 1 0 0 1 483.228 389.254 cm 0 g 0 G 1 0 0 1 -483.228 -389.254 cm BT /F21 8.2889 Tf 483.228 389.254 Td [(].)]TJ ET 1 0 0 1 40.867 389.254 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -389.254 cm BT /F21 8.2889 Tf 52.822 378.805 Td [(Let)-296(us)-297(state)-297(the)-296(basics)-297(of)-296(the)-297(well-known)-296(vertex)-297(operator)-296(construction)-297(for)-296(fundamental)]TJ/F158 9.5641 Tf 338.113 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 8.2889 Tf 4.335 0 Td [(-modules)-297(\050cf.)-296([)]TJ ET 1 0 0 1 450.59 378.805 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -450.59 -378.805 cm BT /F21 8.2889 Tf 450.59 378.805 Td [(27)]TJ ET 1 0 0 1 460.188 378.805 cm 0 g 0 G 1 0 0 1 -460.188 -378.805 cm BT /F21 8.2889 Tf 460.188 378.805 Td [(,)]TJ ET 1 0 0 1 462.128 378.805 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -462.128 -378.805 cm BT /F21 8.2889 Tf 462.128 378.805 Td [(28)]TJ ET 1 0 0 1 471.726 378.805 cm 0 g 0 G 1 0 0 1 -471.726 -378.805 cm BT /F21 8.2889 Tf 471.726 378.805 Td [(]\051.)-297(Define)]TJ ET 1 0 0 1 40.867 378.805 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -378.805 cm BT /F21 8.2889 Tf 40.867 367.345 Td [(Fock)-250(space)]TJ/F108 8.2889 Tf 42.919 0 Td [(M)]TJ/F156 9.5641 Tf 8.053 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(as)-250(induced)]TJ/F158 9.5641 Tf 43.471 1.552 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F21 8.2889 Tf 4.487 0 Td [(-module)]TJ ET 1 0 0 1 40.867 367.345 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -367.345 cm BT /F108 8.2889 Tf 64.777 350.448 Td [(M)]TJ/F156 9.5641 Tf 8.052 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(U)]TJ/F156 9.5641 Tf 6.502 0 Td [(.)]TJ/F158 9.5641 Tf 4.219 1.552 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F156 9.5641 Tf 4.487 0 Td [(/)]TJ/F158 9.5641 Tf 5.161 0 Td [(\012)]TJ/F108 5.8022 Tf 7.651 -1.434 Td [(U)]TJ/F156 6.6948 Tf 4.552 0 Td [(.)]TJ/F80 6.0254 Tf 2.497 0 Td [(h)]TJ/F158 6.6948 Tf 3.141 0 Td [(\012)]TJ/F67 6.0254 Tf 5.356 0 Td [(C)]TJ/F158 6.6948 Tf 4.352 0 Td [(T)]TJ/F108 5.8022 Tf 2.229 0 Td [(t)]TJ/F158 6.6948 Tf 2.55 0 Td [(U\010)]TJ/F67 6.0254 Tf 7.585 0 Td [(C)]TJ/F108 5.8022 Tf 4.352 0 Td [(c)]TJ/F156 6.6948 Tf 3.02 0 Td [(/)]TJ/F67 8.6077 Tf 4.589 1.434 Td [(C)]TJ/F156 9.5641 Tf 6.217 0 Td [(;)]TJ ET 1 0 0 1 40.867 350.448 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -350.448 cm BT /F21 8.2889 Tf 40.867 332.701 Td [(where)]TJ/F158 9.5641 Tf 27.037 1.552 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F158 9.5641 Tf 7.28 0 Td [(D)]TJ/F80 8.6077 Tf 10.253 0 Td [(h)]TJ/F158 9.5641 Tf 6.666 0 Td [(\012)]TJ/F67 8.6077 Tf 9.831 0 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(t)]TJ/F156 9.5641 Tf 3.643 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(t)]TJ/F158 6.6948 Tf 3.642 3.471 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.858 -3.471 Td [(U)-228(\010)]TJ/F67 8.6077 Tf 15.196 0 Td [(C)]TJ/F108 8.2889 Tf 6.216 0 Td [(c)]TJ/F21 8.2889 Tf 4.315 0 Td [(,)-256(such)-257(that)]TJ/F80 8.6077 Tf 41.487 0 Td [(h)]TJ/F158 9.5641 Tf 6.667 0 Td [(\012)]TJ/F67 8.6077 Tf 9.831 0 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.184 0 Td [(t)]TJ/F158 9.5641 Tf 3.643 0 Td [(U)]TJ/F21 8.2889 Tf 5.311 0 Td [(acts)-256(trivially)-257(and)]TJ/F108 8.2889 Tf 65.425 0 Td [(c)]TJ/F21 8.2889 Tf 6.44 0 Td [(as)-256(identity)-257(on)-256(one-dimensional)-257(mod)1(ule)]TJ/F67 8.6077 Tf 152.142 0 Td [(C)]TJ/F21 8.2889 Tf 6.216 0 Td [(.)-256(Denote)-257(by)]TJ ET 1 0 0 1 40.867 332.701 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -332.701 cm BT /F158 9.5641 Tf 40.867 321.241 Td [(f)]TJ/F21 8.2889 Tf 3.538 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.471 Td [(\025)]TJ/F158 9.5641 Tf 7.462 -3.471 Td [(j)]TJ/F156 9.5641 Tf 6.037 0 Td [(\025)]TJ/F158 9.5641 Tf 8.533 0 Td [(2)]TJ/F108 8.2889 Tf 9.317 0 Td [(P)]TJ/F158 9.5641 Tf 5.243 0 Td [(g)]TJ/F21 8.2889 Tf 5.862 0 Td [(a)-280(basis)-281(of)-280(the)-280(group)-281(algebra)]TJ/F67 8.6077 Tf 107.789 0 Td [(C)]TJ/F158 9.5641 Tf 6.216 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(P)]TJ/F158 9.5641 Tf 5.243 0 Td [(U)]TJ/F21 8.2889 Tf 5.508 0 Td [(of)]TJ/F108 8.2889 Tf 9.784 0 Td [(P)]TJ/F21 8.2889 Tf 5.243 0 Td [(.)-280(Then)]TJ/F108 8.2889 Tf 25.628 0 Td [(M)]TJ/F156 9.5641 Tf 8.052 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 9.5641 Tf 5.95 0 Td [(\012)]TJ/F67 8.6077 Tf 10.035 0 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(P)]TJ/F158 9.5641 Tf 5.242 0 Td [(U)]TJ/F21 8.2889 Tf 5.509 0 Td [(is)-280(an)]TJ/F158 9.5641 Tf 20.798 1.552 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F21 8.2889 Tf 4.487 0 Td [(-module:)]TJ/F158 9.5641 Tf 38.435 1.552 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F67 6.0254 Tf 4.487 -1.435 Td [(Z)]TJ/F158 9.5641 Tf 7.817 1.435 Td [(D)]TJ/F159 9.5641 Tf 10.761 7.173 Td [(`)]TJ/F108 5.8022 Tf 8.99 -9.899 Td [(n)]TJ/F158 6.6948 Tf 3.354 0 Td [(2)]TJ/F67 6.0254 Tf 4.211 0 Td [(Z)]TJ/F158 6.6948 Tf 4.017 0 Td [(nf)]TJ/F21 5.8022 Tf 5.724 0 Td [(0)]TJ/F158 6.6948 Tf 3.359 0 Td [(g)]TJ/F80 8.6077 Tf 4.57 2.726 Td [(h)]TJ/F158 9.5641 Tf 6.87 0 Td [(\012)]TJ/F108 8.2889 Tf 10.034 0 Td [(t)]TJ/F108 5.8022 Tf 3.643 3.472 Td [(n)]TJ/F158 9.5641 Tf 6.235 -3.472 Td [(\010)]TJ/F67 8.6077 Tf 10.035 0 Td [(C)]TJ/F108 8.2889 Tf 6.216 0 Td [(c)]TJ/F21 8.2889 Tf 6.638 0 Td [(acts)-280(as)]TJ ET 1 0 0 1 40.867 321.241 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -321.241 cm BT /F158 9.5641 Tf 41.518 308.996 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F67 6.0254 Tf 4.486 -1.434 Td [(Z)]TJ/F158 9.5641 Tf 6.641 1.434 Td [(\012)]TJ/F21 8.2889 Tf 9.776 0 Td [(1,)-250(and)]TJ/F80 8.6077 Tf 25.132 0 Td [(h)]TJ/F158 9.5641 Tf 7.144 0 Td [(D)]TJ/F80 8.6077 Tf 10.116 0 Td [(h)]TJ/F158 9.5641 Tf 6.612 0 Td [(\012)]TJ/F108 8.2889 Tf 9.777 0 Td [(t)]TJ/F21 5.8022 Tf 3.643 3.472 Td [(0)]TJ/F21 8.2889 Tf 5.93 -3.472 Td [(as)-250(1)]TJ/F158 9.5641 Tf 16.987 0 Td [(\012)]TJ/F80 8.6077 Tf 9.777 0 Td [(h)]TJ/F21 8.2889 Tf 4.486 0 Td [(,)-250(with)]TJ/F108 8.2889 Tf 23.806 0 Td [(h)]TJ/F156 9.5641 Tf 4.675 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(0)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(given)-250(by)]TJ/F108 8.2889 Tf 34.341 0 Td [(h)]TJ/F156 9.5641 Tf 4.675 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(/:)]TJ/F21 8.2889 Tf 6.198 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.472 Td [(\025)]TJ/F158 9.5641 Tf 6.817 -3.472 Td [(D)-278(h)]TJ/F108 8.2889 Tf 13.626 0 Td [(h)]TJ/F156 9.5641 Tf 4.675 0 Td [(;)-167(\025)]TJ/F158 9.5641 Tf 9.992 0 Td [(i)]TJ/F21 8.2889 Tf 3.51 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.472 Td [(\025)]TJ/F21 8.2889 Tf 6.232 -3.472 Td [(for)]TJ/F108 8.2889 Tf 12.931 0 Td [(h)]TJ/F158 9.5641 Tf 7.331 0 Td [(2)]TJ/F80 8.6077 Tf 8.673 0 Td [(h)]TJ/F156 9.5641 Tf 4.486 0 Td [(;)-167(\025)]TJ/F158 9.5641 Tf 12.648 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(P)]TJ/F21 8.2889 Tf 5.242 0 Td [(.)]TJ ET 1 0 0 1 40.867 307.444 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -307.444 cm BT /F21 8.2889 Tf 52.822 296.995 Td [(For)]TJ/F156 9.5641 Tf 14.389 0 Td [(\025)]TJ/F158 9.5641 Tf 7.888 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(P)]TJ/F21 8.2889 Tf 7.315 0 Td [(we)-250(have)-250(vertex)-250(operators:)]TJ ET 1 0 0 1 40.867 296.995 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -296.995 cm BT /F108 8.2889 Tf 64.777 278.453 Td [(Y)]TJ/F156 9.5641 Tf 5.864 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\025)]TJ/F156 9.5641 Tf 4.16 -3.95 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(E)]TJ/F158 6.6948 Tf 5.044 3.95 Td [(\000)]TJ/F156 9.5641 Tf 5.72 -3.95 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F156 9.5641 Tf 7.46 0 Td [(\025;)]TJ/F108 8.2889 Tf 9.991 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(E)]TJ/F158 6.6948 Tf 5.044 3.95 Td [(C)]TJ/F156 9.5641 Tf 5.72 -3.95 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F156 9.5641 Tf 7.46 0 Td [(\025;)]TJ/F108 8.2889 Tf 9.991 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/\017)]TJ/F156 6.6948 Tf 7.747 -1.434 Td [(\025)]TJ/F21 8.2889 Tf 4.161 1.434 Td [(e)]TJ/F156 6.6948 Tf 4.359 3.95 Td [(\025)]TJ/F108 8.2889 Tf 4.161 -3.95 Td [(z)]TJ/F156 6.6948 Tf 4.264 3.95 Td [(\025)]TJ/F156 9.5641 Tf 4.16 -3.95 Td [(;)]TJ/F21 8.2889 Tf 276.839 0 Td [(\0504.1\051)]TJ ET 1 0 0 1 40.867 278.453 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -278.453 cm BT /F21 8.2889 Tf 40.867 262.556 Td [(with)]TJ ET 1 0 0 1 40.867 262.556 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -262.556 cm BT /F108 8.2889 Tf 64.777 239.312 Td [(E)]TJ/F158 6.6948 Tf 5.044 3.949 Td [(\006)]TJ/F156 9.5641 Tf 5.72 -3.949 Td [(.\025;)]TJ/F108 8.2889 Tf 13.558 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(exp)]TJ/F159 9.5641 Tf 15.511 16.297 Td [(\040)]TJ 6.628 -7.212 Td [(X)]TJ/F108 5.8022 Tf 0.886 -19.542 Td [(n)]TJ/F158 6.6948 Tf 3.353 0 Td [(\025)]TJ/F21 5.8022 Tf 5.048 0 Td [(1)]TJ/F156 9.5641 Tf 5.84 10.457 Td [(\025.)]TJ/F158 9.5641 Tf 8.799 0 Td [(\006)]TJ/F108 8.2889 Tf 7.46 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F108 8.2889 Tf 4.763 6.704 Td [(z)]TJ/F158 6.6948 Tf 4.264 3.472 Td [(\007)]TJ/F108 5.8022 Tf 5.222 0 Td [(n)]TJ ET q 1 0 0 1 172.784 241.703 cm []0 d 0 J 0.44 w 0 0 m 13.338 0 l S Q BT /F158 9.5641 Tf 173.327 232.521 Td [(\006)]TJ/F108 8.2889 Tf 7.46 0 Td [(n)]TJ/F159 9.5641 Tf 6.53 23.088 Td [(!)]TJ/F156 9.5641 Tf 8.222 -16.297 Td [(;)-1033(\025)]TJ/F158 9.5641 Tf 20.937 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(P)]TJ ET 1 0 0 1 40.867 239.312 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -239.312 cm BT /F156 9.5641 Tf 64.777 214.541 Td [(\017)]TJ/F156 6.6948 Tf 4.179 -1.434 Td [(\025)]TJ/F21 8.2889 Tf 4.161 1.434 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\025)]TJ/F156 9.5641 Tf 4.16 -3.95 Td [(.v)]TJ/F158 9.5641 Tf 10.733 0 Td [(\012)]TJ/F21 8.2889 Tf 9.776 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\026)]TJ/F156 9.5641 Tf 5.091 -3.95 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\017)-55(.\025;)-167(\025)]TJ/F158 9.5641 Tf 25.621 0 Td [(C)]TJ/F156 9.5641 Tf 9.585 0 Td [(\026/v)]TJ/F158 9.5641 Tf 17.294 0 Td [(\012)]TJ/F21 8.2889 Tf 9.777 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\025)]TJ/F158 6.6948 Tf 3.662 0 Td [(C)]TJ/F156 6.6948 Tf 5.222 0 Td [(\026)]TJ ET 1 0 0 1 40.867 214.541 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -214.541 cm BT /F108 8.2889 Tf 64.777 201.092 Td [(z)]TJ/F156 6.6948 Tf 4.264 3.95 Td [(\025)]TJ/F156 9.5641 Tf 4.161 -3.95 Td [(.v)]TJ/F158 9.5641 Tf 10.733 0 Td [(\012)]TJ/F21 8.2889 Tf 9.776 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\026)]TJ/F156 9.5641 Tf 5.091 -3.95 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(z)]TJ/F158 6.6948 Tf 4.264 3.95 Td [(h)]TJ/F156 6.6948 Tf 2.457 0 Td [(\025;\026)]TJ/F158 6.6948 Tf 10.471 0 Td [(i)]TJ/F156 9.5641 Tf 2.955 -3.95 Td [(v)]TJ/F158 9.5641 Tf 7.166 0 Td [(\012)]TJ/F21 8.2889 Tf 9.776 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\026)]TJ/F156 9.5641 Tf 5.091 -3.95 Td [(;)-1033(v)]TJ/F158 9.5641 Tf 20.746 0 Td [(2)]TJ/F108 8.2889 Tf 8.672 0 Td [(M)]TJ/F156 9.5641 Tf 8.053 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/;)-167(\025;)-166(\026)]TJ/F158 9.5641 Tf 27.536 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(P)]TJ/F156 9.5641 Tf 5.242 0 Td [(;)]TJ ET 1 0 0 1 40.867 201.092 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -201.092 cm BT /F21 8.2889 Tf 40.867 185.194 Td [(where)]TJ/F156 9.5641 Tf 26.333 0 Td [(\017)]TJ/F21 8.2889 Tf 6.778 0 Td [(is)-250(a)-250(2-cocyle)-250(corresponding)-250(to)-250(a)-250(central)-250(extension)-250(of)]TJ/F108 8.2889 Tf 199.993 0 Td [(P)]TJ/F21 8.2889 Tf 7.315 0 Td [(by)-250(certain)-250(finite)-250(cyclic)-250(group)-250(\050cf.)-250([)]TJ ET 1 0 0 1 408.578 185.194 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -408.578 -185.194 cm BT /F21 8.2889 Tf 408.578 185.194 Td [(23)]TJ ET 1 0 0 1 418.176 185.194 cm 0 g 0 G 1 0 0 1 -418.176 -185.194 cm BT /F21 8.2889 Tf 418.176 185.194 Td [(,)]TJ ET 1 0 0 1 420.116 185.194 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -420.116 -185.194 cm BT /F21 8.2889 Tf 420.116 185.194 Td [(21)]TJ ET 1 0 0 1 429.714 185.194 cm 0 g 0 G 1 0 0 1 -429.714 -185.194 cm BT /F21 8.2889 Tf 429.714 185.194 Td [(]\051.)]TJ ET 1 0 0 1 40.867 185.194 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -185.194 cm BT /F21 8.2889 Tf 52.822 173.746 Td [(The)-251(action)-250(of)]TJ/F158 9.5641 Tf 51.915 1.552 Td [(O)]TJ/F80 8.6077 Tf -0.651 -1.552 Td [(h)]TJ/F21 8.2889 Tf 6.564 0 Td [(extends)-251(to)-250(the)-251(action)-250(of)]TJ/F158 9.5641 Tf 92.316 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 8.2889 Tf 6.413 0 Td [(via)]TJ ET 1 0 0 1 222.154 173.746 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -222.154 -173.746 cm BT /F21 8.2889 Tf 222.154 173.746 Td [(\0504.1\051)]TJ ET 1 0 0 1 239.826 173.746 cm 0 g 0 G 1 0 0 1 -239.826 -173.746 cm BT /F21 8.2889 Tf 241.903 173.746 Td [(in)-251(the)-250(following)-251(manner:)]TJ/F108 8.2889 Tf 97.231 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\013)]TJ/F156 9.5641 Tf 4.589 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F21 8.2889 Tf 5.644 0 Td [(acts)-251(on)]TJ/F108 8.2889 Tf 28.93 0 Td [(M)]TJ/F156 9.5641 Tf 8.053 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 9.5641 Tf 5.698 0 Td [(\012)]TJ/F67 8.6077 Tf 9.782 0 Td [(C)]TJ/F158 9.5641 Tf 6.216 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(P)]TJ/F158 9.5641 Tf 5.243 0 Td [(U)]TJ/F21 8.2889 Tf 5.262 0 Td [(as)-251(the)-250(coefficient)]TJ ET 1 0 0 1 40.867 173.746 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -173.746 cm BT /F21 8.2889 Tf 40.867 163.286 Td [(of)]TJ/F108 8.2889 Tf 9.532 0 Td [(x)]TJ/F158 6.6948 Tf 4.07 3.471 Td [(\000)]TJ/F108 5.8022 Tf 5.222 0 Td [(n)]TJ/F158 6.6948 Tf 3.353 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 5.93 -3.471 Td [(in)]TJ/F108 8.2889 Tf 9.582 0 Td [(Y)]TJ/F156 9.5641 Tf 5.864 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.471 Td [(\013)]TJ/F156 9.5641 Tf 4.588 -3.471 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(:)]TJ ET 1 0 0 1 40.867 163.286 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -163.286 cm BT /F108 8.2889 Tf 64.777 147.388 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.434 Td [(\013)]TJ/F156 9.5641 Tf 4.588 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(Y)]TJ/F156 9.5641 Tf 5.864 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\013)]TJ/F156 9.5641 Tf 4.588 -3.95 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/;)]TJ ET 1 0 0 1 40.867 147.388 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -147.388 cm BT /F21 8.2889 Tf 40.867 131.491 Td [(and)]TJ/F108 8.2889 Tf 15.842 0 Td [(d)]TJ/F21 8.2889 Tf 6.202 0 Td [(as)-192(degree)-193(operator.)-192(Furthermore,)-193(the)-192(direct)-192(summands)]TJ/F108 8.2889 Tf 207.158 0 Td [(M)]TJ/F156 9.5641 Tf 8.053 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(/)]TJ/F158 9.5641 Tf 4.22 0 Td [(\012)]TJ/F21 8.2889 Tf 8.305 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.472 Td [(!)]TJ/F108 4.9734 Tf 4.385 -1.352 Td [(i)]TJ/F67 8.6077 Tf 2.419 -2.12 Td [(C)]TJ/F158 9.5641 Tf 6.216 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(Q)]TJ/F158 9.5641 Tf 6.934 0 Td [(U)]TJ/F21 8.2889 Tf 4.778 0 Td [(of)]TJ/F108 8.2889 Tf 9.054 0 Td [(M)]TJ/F156 9.5641 Tf 8.053 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 9.5641 Tf 4.221 0 Td [(\012)]TJ/F67 8.6077 Tf 8.305 0 Td [(C)]TJ/F158 9.5641 Tf 6.216 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(P)]TJ/F158 9.5641 Tf 5.243 0 Td [(U)]TJ/F21 8.2889 Tf 4.779 0 Td [(are)-192(exactly)-192(standard)]TJ/F158 9.5641 Tf 78.406 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 8.2889 Tf 4.335 0 Td [(-modules)]TJ ET 1 0 0 1 40.867 131.491 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -131.491 cm BT /F108 8.2889 Tf 40.867 121.03 Td [(L)]TJ/F156 9.5641 Tf 3.928 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F108 5.8022 Tf 7.327 -1.434 Td [(i)]TJ/F156 9.5641 Tf 2.158 1.434 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)-250(and)-250(we)-250(can)-250(identify)-250(highest)-250(weight)-250(vectors)]TJ/F156 9.5641 Tf 169.673 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.434 Td [(\003)]TJ/F108 4.9734 Tf 5.129 -1.352 Td [(i)]TJ/F158 9.5641 Tf 5.075 2.786 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F158 9.5641 Tf 6.925 0 Td [(\012)]TJ/F21 8.2889 Tf 9.776 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.472 Td [(!)]TJ/F108 4.9734 Tf 4.385 -1.352 Td [(i)]TJ/F21 8.2889 Tf 2.419 -2.12 Td [(,)]TJ/F108 8.2889 Tf 4.012 0 Td [(i)]TJ/F158 9.5641 Tf 5.027 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F21 8.2889 Tf 4.759 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F21 8.2889 Tf 26.667 0 Td [(.)]TJ ET 1 0 0 1 40.867 121.03 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -121.03 cm BT /F21 8.2889 Tf 52.822 110.581 Td [(For)]TJ/F156 9.5641 Tf 14.389 0 Td [(\025)]TJ/F158 9.5641 Tf 7.888 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(P)]TJ/F21 8.2889 Tf 7.315 0 Td [(we)-250(also)-250(have)-250(Dong\025Lepowsky's)-250(intertwining)-250(operators:)]TJ ET 1 0 0 1 40.867 110.581 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -110.581 cm BT /F193 8.9265 Tf 64.777 94.684 Td [(Y)]TJ/F156 9.5641 Tf 6.641 0 Td [(.)]TJ/F21 8.2889 Tf 3.568 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.95 Td [(\025)]TJ/F156 9.5641 Tf 4.16 -3.95 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(V)56(D)]TJ/F108 8.2889 Tf 12.216 0 Td [(E)]TJ/F158 6.6948 Tf 5.043 3.95 Td [(\000)]TJ/F156 9.5641 Tf 5.72 -3.95 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F156 9.5641 Tf 7.46 0 Td [(\025;)]TJ/F108 8.2889 Tf 9.991 0 Td [(z)]TJ/F156 9.5641 Tf 4.265 0 Td [(/)]TJ/F108 8.2889 Tf 3.567 0 Td [(E)]TJ/F158 6.6948 Tf 5.044 3.95 Td [(C)]TJ/F156 9.5641 Tf 5.72 -3.95 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\000)]TJ/F156 9.5641 Tf 7.46 0 Td [(\025;)]TJ/F108 8.2889 Tf 9.992 0 Td [(z)]TJ/F156 9.5641 Tf 4.264 0 Td [(/)]TJ/F108 8.2889 Tf 3.568 0 Td [(e)]TJ/F156 6.6948 Tf 3.978 -1.434 Td [(\025)]TJ/F108 8.2889 Tf 4.16 1.434 Td [(z)]TJ/F156 6.6948 Tf 4.265 3.95 Td [(\025)]TJ/F21 8.2889 Tf 4.16 -3.95 Td [(e)]TJ/F21 5.8022 Tf 4.36 3.95 Td [(i)]TJ/F156 6.6948 Tf 1.741 0 Td [(\031)-100(\025)]TJ/F108 8.2889 Tf 8.846 -3.95 Td [(c)]TJ/F156 9.5641 Tf 4.315 0 Td [(.)]TJ/F158 9.5641 Tf 3.567 0 Td [(\001)]TJ/F156 9.5641 Tf 2.659 0 Td [(;)-167(\025/;)]TJ ET 1 0 0 1 40.867 94.684 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -94.684 cm BT /F21 8.2889 Tf 40.867 78.787 Td [(where)]TJ/F108 8.2889 Tf 26.333 0 Td [(e)]TJ/F156 6.6948 Tf 3.979 -1.435 Td [(\025)]TJ/F158 9.5641 Tf 6.817 1.435 Td [(D)]TJ/F21 8.2889 Tf 10.116 0 Td [(e)]TJ/F156 6.6948 Tf 4.36 3.472 Td [(\025)]TJ/F156 9.5641 Tf 4.16 -3.472 Td [(\017)-55(.\025;)]TJ/F158 9.5641 Tf 18.265 0 Td [(\001)]TJ/F156 9.5641 Tf 2.659 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F108 8.2889 Tf 4.012 0 Td [(c)]TJ/F156 9.5641 Tf 4.314 0 Td [(.\013)-40(;)-167(\014)-70(/)]TJ/F21 8.2889 Tf 25.874 0 Td [(defined)-250(in)-250([)]TJ ET 1 0 0 1 198.823 78.787 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -198.823 -78.787 cm BT /F21 8.2889 Tf 198.823 78.787 Td [(21)]TJ ET 1 0 0 1 208.421 78.787 cm 0 g 0 G 1 0 0 1 -208.421 -78.787 cm BT /F21 8.2889 Tf 208.421 78.787 Td [(],)-250(Eq.)-250(\05012.52\051.)]TJ ET 1 0 0 1 40.867 78.787 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 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Tf 9.777 0 Td [(t)]TJ/F108 5.8022 Tf 3.643 3.472 Td [(n)]TJ/F21 8.2889 Tf 3.852 -3.472 Td [(.)]TJ ET 1 0 0 1 37.134 642.588 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -642.588 cm BT /F21 8.2889 Tf 49.089 631.141 Td [(We)-250(denote)]TJ/F108 8.2889 Tf 43.608 0 Td [(x)]TJ/F156 9.5641 Tf 4.07 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(x)]TJ/F158 9.5641 Tf 6.195 0 Td [(\012)]TJ/F108 8.2889 Tf 9.776 0 Td [(t)]TJ/F108 5.8022 Tf 3.643 3.472 Td [(n)]TJ/F21 8.2889 Tf 5.924 -3.472 Td [(for)]TJ/F108 8.2889 Tf 12.931 0 Td [(x)]TJ/F158 9.5641 Tf 6.726 0 Td [(2)]TJ/F80 8.6077 Tf 8.673 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(,)]TJ/F108 8.2889 Tf 4.011 0 Td [(n)]TJ/F158 9.5641 Tf 7.448 0 Td [(2)]TJ/F67 8.6077 Tf 8.672 0 Td [(Z)]TJ/F21 8.2889 Tf 5.739 0 Td [(,)-250(and)-250(define)-250(the)-250(following)-250(formal)-250(Laurent)-250(series)-250(in)-250(formal)-250(variable)]TJ/F108 8.2889 Tf 251.641 0 Td 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4.486 3.472 Td [(e)]TJ/F158 9.5641 Tf 5.94 -3.472 Td [(D)]TJ/F80 8.6077 Tf 10.117 0 Td [(h)]TJ/F158 9.5641 Tf 6.202 0 Td [(\010)]TJ/F67 8.6077 Tf 9.367 0 Td [(C)]TJ/F108 8.2889 Tf 6.217 0 Td [(c)]TJ/F158 9.5641 Tf 6.03 0 Td [(\010)]TJ/F67 8.6077 Tf 9.367 0 Td [(C)]TJ/F108 8.2889 Tf 6.216 0 Td [(d)]TJ/F21 8.2889 Tf 4.609 0 Td [(,)]TJ/F158 9.5641 Tf 4.553 0 Td [(Q)]TJ/F80 8.6077 Tf -0.674 0 Td [(n)]TJ/F158 6.6948 Tf 4.533 -1.435 Td [(\006)]TJ/F158 9.5641 Tf 8.377 1.435 Td [(D)]TJ/F80 8.6077 Tf 10.116 0 Td [(g)]TJ/F158 9.5641 Tf 6.052 0 Td [(\012)]TJ/F108 8.2889 Tf 9.367 0 Td [(t)]TJ/F158 6.6948 Tf 3.643 3.472 Td [(\006)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F67 8.6077 Tf 3.857 -3.472 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(t)]TJ/F158 6.6948 Tf 3.643 3.472 Td [(\006)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.857 -3.472 Td [(U)-179(\010)]TJ/F80 8.6077 Tf 14.267 0 Td [(n)]TJ/F158 6.6948 Tf 4.534 -1.435 Td [(\006)]TJ/F21 8.2889 Tf 5.72 1.435 Td [(,)-234(we)-234(have)-234(the)-234(triangular)-234(decomposition)-234(for)]TJ/F158 9.5641 Tf 163.796 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(:)]TJ ET 1 0 0 1 37.134 580.963 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -580.963 cm BT /F158 9.5641 Tf 61.62 561.964 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F158 9.5641 Tf 6.993 0 Td [(D)-348(Q)]TJ/F80 8.6077 Tf 10.116 0 Td [(n)]TJ/F158 6.6948 Tf 4.534 -1.434 Td [(\000)]TJ/F158 9.5641 Tf 7.845 1.434 Td [(\010)]TJ/F80 8.6077 Tf 9.777 0 Td [(h)]TJ/F108 5.8022 Tf 4.486 3.95 Td [(e)]TJ/F158 9.5641 Tf 5.409 -3.95 Td [(\010)-293(Q)]TJ/F80 8.6077 Tf 9.777 0 Td [(n)]TJ/F158 6.6948 Tf 4.533 -1.434 Td [(C)]TJ/F156 9.5641 Tf 5.72 1.434 Td [(:)]TJ ET 1 0 0 1 37.134 561.964 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -561.964 cm BT /F21 8.2889 Tf 37.134 542.966 Td [(As)-315(usual,)-315(denote)-314(by)]TJ/F158 9.5641 Tf 77.784 0 Td [(f)]TJ/F156 9.5641 Tf 3.538 0 Td [(\013)]TJ/F21 5.8022 Tf 5.461 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(\013)]TJ/F21 5.8022 Tf 10.221 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(\013)]TJ/F156 6.6948 Tf 27.653 -1.435 Td [(`)]TJ/F158 9.5641 Tf 3.632 1.435 Td [(g)-422(\032)]TJ/F156 9.5641 Tf 19.065 0 Td [(.)]TJ/F80 8.6077 Tf 3.567 0 Td [(h)]TJ/F108 5.8022 Tf 4.487 3.471 Td [(e)]TJ/F156 9.5641 Tf 3.283 -3.471 Td [(/)]TJ/F158 6.6948 Tf 3.568 3.471 Td [(\003)]TJ/F21 8.2889 Tf 6.595 -3.471 Td [(the)-315(corresponding)-314(set)-315(of)-315(simple)-315(roots,)-315(and)-314(by)]TJ/F156 9.2055 Tf 177.872 0 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)]TJ/F156 9.2055 Tf 4.76 0 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F156 9.2055 Tf 22.192 0 Td [(\003)]TJ/F156 6.6948 Tf 7.327 -1.435 Td [(`)]TJ/F21 8.2889 Tf 6.241 1.435 Td [(fundamental)]TJ ET 1 0 0 1 37.134 542.966 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -542.966 cm BT /F21 8.2889 Tf 37.134 532.505 Td [(weights.)]TJ ET 1 0 0 1 37.134 532.505 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -532.505 cm BT /F21 8.2889 Tf 49.089 521.058 Td [(Let)]TJ/F108 8.2889 Tf 13.66 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(denote)-250(standard)-250(\050i.e.,)-250(integrable)-250(highest)-250(weight\051)]TJ/F158 9.5641 Tf 185.474 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 8.2889 Tf 4.335 0 Td [(-module)-250(with)-250(dominant)-250(integral)-250(highest)-250(weight)]TJ ET 1 0 0 1 37.134 521.058 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -521.058 cm BT /F156 9.2055 Tf 61.044 502.059 Td [(\003)]TJ/F158 9.5641 Tf 9.984 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.586 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(1)]TJ/F156 9.2055 Tf 3.857 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.632 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.327 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.632 1.435 Td [(;)]TJ/F108 8.2889 Tf 13.048 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.435 Td [(2)]TJ/F67 8.6077 Tf 8.673 0 Td [(Z)]TJ/F158 6.6948 Tf 5.738 -1.435 Td [(C)]TJ/F156 9.5641 Tf 5.72 1.435 Td [(:)]TJ ET 1 0 0 1 37.134 502.059 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -502.059 cm BT /F21 8.2889 Tf 37.134 483.06 Td [(Define)-250(the)-250(level)-250(of)]TJ/F108 8.2889 Tf 71.325 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 9.5641 Tf 7.327 0 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(as)]TJ/F108 8.2889 Tf 10.063 0 Td [(k)]TJ/F158 9.5641 Tf 6.966 0 Td [(D)]TJ/F156 9.2055 Tf 10.117 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(c)]TJ/F156 9.5641 Tf 4.314 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)]TJ/F108 8.2889 Tf 9.585 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.434 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-222(\001)-167(\001)-166(\001)-223(C)]TJ/F108 8.2889 Tf 32.46 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.434 Td [(`)]TJ/F21 8.2889 Tf 3.631 1.434 Td [(.)]TJ ET 1 0 0 1 37.134 483.06 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -483.06 cm BT /F123 8.2889 Tf 37.134 453.073 Td [(3.)-500(Feigin\025Stoyanovsky's)-250(type)-250(subspaces)]TJ ET 1 0 0 1 37.134 453.073 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -453.073 cm BT /F108 8.2889 Tf 37.134 422.999 Td [(3.1.)-500(Definition)]TJ ET 1 0 0 1 37.134 422.999 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -422.999 cm BT /F21 8.2889 Tf 49.089 398.133 Td [(Fix)-250(the)-250(minuscule)-250(weight)]TJ/F156 9.5641 Tf 98.306 0 Td [(!)]TJ/F158 9.5641 Tf 9.112 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(!)]TJ/F156 6.6948 Tf 6.264 -1.435 Td [(`)]TJ/F21 8.2889 Tf 5.704 1.435 Td [(and)-250(define)-250(the)-250(following)-250(alternative)-250(basis)-250(for)]TJ/F80 8.6077 Tf 172.59 0 Td [(h)]TJ/F158 6.6948 Tf 4.487 3.472 Td [(\003)]TJ/F21 8.2889 Tf 3.986 -3.472 Td [(:)]TJ ET 1 0 0 1 37.134 398.133 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -398.133 cm BT /F156 9.2055 Tf 61.044 379.134 Td [(\000)]TJ/F158 9.5641 Tf 10.224 0 Td [(D)-278(f)]TJ/F156 9.5641 Tf 13.655 0 Td [(\013)]TJ/F158 9.5641 Tf 8.5 0 Td [(2)]TJ/F108 8.2889 Tf 8.673 0 Td [(R)]TJ/F158 9.5641 Tf 7.522 0 Td [(j)]TJ/F156 9.5641 Tf 5.392 0 Td [(!)-20(.\013)-40(/)]TJ/F158 9.5641 Tf 22.091 0 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)-278(D)-278(f)]TJ/F156 9.5641 Tf 19.851 0 Td [(\015)]TJ/F21 5.8022 Tf 4.801 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(\015)]TJ/F21 5.8022 Tf 9.561 -1.435 Td [(2)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(\015)]TJ/F156 6.6948 Tf 26.993 -1.435 Td [(`)]TJ/F158 9.5641 Tf 6.288 1.435 Td [(j)]TJ/F156 9.5641 Tf 5.392 0 Td [(\015)]TJ/F108 5.8022 Tf 4.801 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.814 1.435 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\017)]TJ/F108 5.8022 Tf 4.179 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.283 1.435 Td [(\000)]TJ/F156 9.5641 Tf 9.586 0 Td [(\017)]TJ/F156 6.6948 Tf 4.179 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(\013)]TJ/F108 5.8022 Tf 5.461 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.283 1.435 Td [(C)-222(\001)-167(\001)-167(\001)-222(C)]TJ/F156 9.5641 Tf 32.461 0 Td [(\013)]TJ/F156 6.6948 Tf 5.461 -1.435 Td [(`)]TJ/F158 9.5641 Tf 3.631 1.435 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(:)]TJ ET 1 0 0 1 37.134 379.134 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -379.134 cm BT /F21 8.2889 Tf 37.134 360.135 Td [(Consequently)-250(we)-250(obtain)]TJ/F67 8.6077 Tf 94.194 0 Td [(Z)]TJ/F21 8.2889 Tf 5.739 0 Td [(-grading)-250(of)]TJ/F80 8.6077 Tf 44.13 0 Td [(g)]TJ/F21 8.2889 Tf 4.335 0 Td [(:)]TJ ET 1 0 0 1 37.134 360.135 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -360.135 cm BT /F80 8.6077 Tf 61.044 341.137 Td [(g)]TJ/F158 9.5641 Tf 6.993 0 Td [(D)]TJ/F80 8.6077 Tf 10.116 0 Td [(g)]TJ/F158 6.6948 Tf 4.336 -1.435 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F80 8.6077 Tf 9.585 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(C)]TJ/F80 8.6077 Tf 9.585 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)]TJ ET 1 0 0 1 37.134 341.137 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -341.137 cm BT /F21 8.2889 Tf 37.134 322.138 Td [(where)]TJ/F80 8.6077 Tf 26.334 0 Td [(g)]TJ/F21 5.8022 Tf 4.335 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.515 1.435 Td [(D)]TJ/F80 8.6077 Tf 10.116 0 Td [(h)]TJ/F158 9.5641 Tf 6.612 0 Td [(C)]TJ/F159 9.5641 Tf 9.586 7.173 Td [(P)]TJ/F156 6.6948 Tf 9.994 -9.899 Td [(!)-20(.\013)-40(/)]TJ/F158 6.6948 Tf 13.604 0 Td [(D)]TJ/F21 5.8022 Tf 5.222 0 Td [(0)]TJ/F80 8.6077 Tf 5.452 2.726 Td [(g)]TJ/F156 6.6948 Tf 4.335 -1.435 Td [(\013)]TJ/F21 8.2889 Tf 4.589 1.435 Td [(,)]TJ/F80 8.6077 Tf 4.012 0 Td [(g)]TJ/F158 6.6948 Tf 4.335 -1.435 Td [(\006)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.515 1.435 Td [(D)]TJ/F159 9.5641 Tf 10.116 7.173 Td [(P)]TJ/F156 6.6948 Tf 9.995 -9.899 Td [(\013)]TJ/F158 6.6948 Tf 4.09 0 Td [(2\006)]TJ/F156 6.4438 Tf 9.433 0 Td [(\000)]TJ/F80 8.6077 Tf 7.389 2.726 Td [(g)]TJ/F156 6.6948 Tf 4.336 -1.435 Td [(\013)]TJ/F21 8.2889 Tf 4.589 1.435 Td [(.)-250(The)-250(corresponding)]TJ/F67 8.6077 Tf 77.31 0 Td [(Z)]TJ/F21 8.2889 Tf 5.738 0 Td [(-grading)-250(on)-250(affine)-250(Lie)-250(algebra)]TJ/F158 9.5641 Tf 113.967 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 8.2889 Tf 6.408 0 Td [(is)]TJ ET 1 0 0 1 37.134 322.138 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -322.138 cm BT /F158 9.5641 Tf 61.62 301.798 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F158 9.5641 Tf 6.993 0 Td [(D)-338(Q)]TJ/F80 8.6077 Tf 10.116 0 Td [(g)]TJ/F158 6.6948 Tf 4.336 -1.434 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-282(Q)]TJ/F80 8.6077 Tf 9.585 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.434 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.434 Td [(C)-282(Q)]TJ/F80 8.6077 Tf 9.585 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.434 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.434 Td [(;)]TJ ET 1 0 0 1 37.134 301.798 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -301.798 cm BT /F21 8.2889 Tf 37.134 282.8 Td [(with)]TJ/F158 9.5641 Tf 20.369 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 5.8022 Tf 4.335 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.515 1.435 Td [(D)]TJ/F80 8.6077 Tf 10.116 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(\012)]TJ/F67 8.6077 Tf 9.777 0 Td [(C)]TJ/F158 9.5641 Tf 6.216 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(t)]TJ/F156 9.5641 Tf 3.643 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(t)]TJ/F158 6.6948 Tf 3.643 3.472 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.857 -3.472 Td [(U)-222(\010)]TJ/F67 8.6077 Tf 15.087 0 Td [(C)]TJ/F108 8.2889 Tf 6.217 0 Td [(c)]TJ/F158 9.5641 Tf 6.439 0 Td [(\010)]TJ/F67 8.6077 Tf 9.777 0 Td [(C)]TJ/F108 8.2889 Tf 6.217 0 Td [(d)]TJ/F21 8.2889 Tf 4.608 0 Td [(,)]TJ/F158 9.5641 Tf 4.588 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F158 6.6948 Tf 4.336 -1.435 Td [(\006)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(D)]TJ/F80 8.6077 Tf 10.117 0 Td [(g)]TJ/F158 6.6948 Tf 4.335 -1.435 Td [(\006)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(\012)]TJ/F67 8.6077 Tf 9.777 0 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.184 0 Td [(t)]TJ/F156 9.5641 Tf 3.643 0 Td [(;)]TJ/F108 8.2889 Tf 4.76 0 Td [(t)]TJ/F158 6.6948 Tf 3.643 3.472 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.858 -3.472 Td [(U)]TJ/F21 8.2889 Tf 3.184 0 Td [(.)-250(Note)-250(that)]TJ ET 1 0 0 1 37.134 282.8 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -282.8 cm BT /F158 9.5641 Tf 61.62 263.801 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(D)]TJ/F21 8.2889 Tf 10.117 0 Td [(span)]TJ/F158 9.5641 Tf 17.97 0 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\015)]TJ/F156 9.5641 Tf 4.93 1.435 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(n)]TJ/F156 9.5641 Tf 4.791 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(j)]TJ/F156 9.5641 Tf 5.392 0 Td [(\015)]TJ/F158 9.5641 Tf 8.988 0 Td [(2)]TJ/F156 9.2055 Tf 8.673 0 Td [(\000)]TJ/F156 9.5641 Tf 7.567 0 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(n)]TJ/F158 9.5641 Tf 7.448 0 Td [(2)]TJ/F67 8.6077 Tf 8.672 0 Td [(Z)]TJ/F158 9.5641 Tf 5.739 0 Td [(g)]TJ ET 1 0 0 1 37.134 263.801 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -263.801 cm BT /F21 8.2889 Tf 37.134 244.802 Td [(is)-250(a)-250(commutative)-250(subalgebra)-250(and)-250(a)]TJ/F158 9.5641 Tf 134.457 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.434 Td [(0)]TJ/F21 8.2889 Tf 3.858 1.434 Td [(-module.)]TJ ET 1 0 0 1 37.134 244.802 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -244.802 cm BT /F21 8.2889 Tf 49.089 233.355 Td [(For)-250(an)-250(integral)-250(dominant)-250(weight)]TJ/F156 9.2055 Tf 125.427 0 Td [(\003)]TJ/F21 8.2889 Tf 9.4 0 Td [(define)-250(Feigin\025Stoyanovsky's)-250(type)-250(subspace)-250(of)]TJ/F108 8.2889 Tf 175.002 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(as)]TJ ET 1 0 0 1 37.134 233.355 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -233.355 cm BT /F108 8.2889 Tf 61.044 214.357 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 9.5641 Tf 7.327 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(U)]TJ/F156 9.5641 Tf 6.503 0 Td [(.)]TJ/F158 9.5641 Tf 4.143 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(\001)]TJ/F156 9.5641 Tf 4.784 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F156 9.5641 Tf 5.627 1.435 Td [(;)]TJ ET 1 0 0 1 37.134 214.357 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -214.357 cm BT /F108 8.2889 Tf 37.134 195.358 Td [(U)]TJ/F156 9.5641 Tf 6.502 0 Td [(.)]TJ/F158 9.5641 Tf 4.143 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(/)]TJ/F21 8.2889 Tf 5.64 0 Td [(denoting)-250(the)-250(universal)-250(enveloping)-250(algebra)-250(for)]TJ/F158 9.5641 Tf 176.149 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F21 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8.2889 Tf 2.982 -3.471 Td [(.)]TJ ET 1 0 0 1 40.867 628.316 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -628.316 cm BT /F21 8.2889 Tf 52.822 617.889 Td [(In)-278(order)-279(to)-278(state)-279(the)-278(main)-279(result)-278(we)-279(need)-278(some)-279(notation.)-278(Denote)]TJ/F80 8.6077 Tf 248.806 0 Td [(g)]TJ/F158 9.5641 Tf 7.598 0 Td [(D)]TJ/F80 8.6077 Tf 10.722 0 Td [(sl)]TJ/F156 9.5641 Tf 6.216 0 Td [(.`)]TJ/F158 9.5641 Tf 10.411 0 Td [(C)]TJ/F21 8.2889 Tf 9.828 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F67 8.6077 Tf 4.76 0 Td [(C)]TJ/F156 9.5641 Tf 6.216 0 Td [(/)]TJ/F21 8.2889 Tf 5.876 0 Td [(and)-278(let)]TJ/F80 8.6077 Tf 28.795 0 Td [(h)]TJ/F21 8.2889 Tf 6.795 0 Td [(be)-279(a)-278(Cartan)-278(subalgebra)-279(of)]TJ/F80 8.6077 Tf 99.138 0 Td [(g)]TJ ET 1 0 0 1 40.867 617.889 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -617.889 cm BT /F21 8.2889 Tf 40.867 607.428 Td [(with)-250(the)-250(corresponding)-250(root)-250(system)]TJ/F108 8.2889 Tf 138.655 0 Td [(R)]TJ/F21 8.2889 Tf 4.866 0 Td [(.)-250(Let)]TJ ET 1 0 0 1 40.867 607.428 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -607.428 cm BT /F80 8.6077 Tf 64.777 590.429 Td [(g)]TJ/F158 9.5641 Tf 6.992 0 Td [(D)]TJ/F80 8.6077 Tf 10.117 0 Td [(h)]TJ/F158 9.5641 Tf 6.612 0 Td [(C)]TJ/F159 9.5641 Tf 9.585 9.085 Td [(X)]TJ/F156 6.6948 Tf 0.913 -19.541 Td [(\013)]TJ/F158 6.6948 Tf 4.091 0 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(R)]TJ/F80 8.6077 Tf 5.913 10.456 Td [(g)]TJ/F156 6.6948 Tf 4.335 -1.435 Td [(\013)]TJ ET 1 0 0 1 40.867 590.429 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -590.429 cm BT /F21 8.2889 Tf 40.867 566.236 Td [(be)-250(the)-250(root)-250(space)-250(decomposition)-250(of)]TJ/F80 8.6077 Tf 135.481 0 Td [(g)]TJ/F21 8.2889 Tf 6.408 0 Td [(with)-250(fixed)-250(root)-250(vectors)]TJ/F108 8.2889 Tf 88.748 0 Td [(x)]TJ/F156 6.6948 Tf 4.07 -1.435 Td [(\013)]TJ/F21 8.2889 Tf 4.589 1.435 Td [(.)-250(Let)]TJ ET 1 0 0 1 40.867 566.236 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -566.236 cm BT /F80 8.6077 Tf 64.777 550.606 Td [(g)]TJ/F158 9.5641 Tf 6.992 0 Td [(D)]TJ/F80 8.6077 Tf 10.117 0 Td [(g)]TJ/F158 6.6948 Tf 4.336 -1.435 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(\010)]TJ/F80 8.6077 Tf 9.776 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(\010)]TJ/F80 8.6077 Tf 9.776 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ ET 1 0 0 1 40.867 550.606 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -550.606 cm BT /F21 8.2889 Tf 40.867 534.976 Td [(be)-250(a)-250(chosen)]TJ/F67 8.6077 Tf 46.442 0 Td [(Z)]TJ/F21 8.2889 Tf 5.738 0 Td [(-grading)-250(such)-250(that)]TJ/F80 8.6077 Tf 71.914 0 Td [(h)]TJ/F158 9.5641 Tf 7.144 0 Td [(\032)]TJ/F80 8.6077 Tf 10.117 0 Td [(g)]TJ/F21 5.8022 Tf 4.335 -1.435 Td [(0)]TJ/F21 8.2889 Tf 3.858 1.435 Td [(.)-250(Grading)-250(on)]TJ/F80 8.6077 Tf 48.349 0 Td [(g)]TJ/F21 8.2889 Tf 6.408 0 Td [(induces)]TJ/F67 8.6077 Tf 31.522 0 Td [(Z)]TJ/F21 8.2889 Tf 5.739 0 Td [(-grading)-250(on)]TJ/F158 9.5641 Tf 46.968 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(:)]TJ ET 1 0 0 1 40.867 534.976 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -534.976 cm BT /F158 9.5641 Tf 65.352 519.346 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F158 9.5641 Tf 6.992 0 Td [(D)-338(Q)]TJ/F80 8.6077 Tf 10.117 0 Td [(g)]TJ/F158 6.6948 Tf 4.336 -1.435 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(\010)-282(Q)]TJ/F80 8.6077 Tf 9.776 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(0)]TJ/F158 9.5641 Tf 5.983 1.435 Td [(\010)-282(Q)]TJ/F80 8.6077 Tf 9.776 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(;)]TJ ET 1 0 0 1 40.867 519.346 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -519.346 cm BT /F21 8.2889 Tf 40.867 503.716 Td [(where)]TJ/F158 9.5641 Tf 26.909 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F158 9.5641 Tf 6.514 1.435 Td [(D)]TJ/F80 8.6077 Tf 10.117 0 Td [(g)]TJ/F21 5.8022 Tf 4.336 -1.435 Td [(1)]TJ/F158 9.5641 Tf 5.982 1.435 Td [(\012)]TJ/F67 8.6077 Tf 9.777 0 Td [(C)]TJ/F158 9.5641 Tf 6.217 0 Td [(T)]TJ/F108 8.2889 Tf 3.185 0 Td [(t)]TJ/F156 9.5641 Tf 3.643 0 Td [(;)]TJ/F108 8.2889 Tf 4.759 0 Td [(t)]TJ/F158 6.6948 Tf 3.643 3.471 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F158 9.5641 Tf 3.858 -3.471 Td [(U)]TJ/F21 8.2889 Tf 5.257 0 Td [(is)-250(a)-250(commutative)-250(Lie)-250(algebra)-250(with)-250(a)-250(basis)-250(given)-250(by)]TJ ET 1 0 0 1 40.867 503.716 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -503.716 cm BT /F158 9.5641 Tf 64.777 488.085 Td [(f)]TJ/F108 8.2889 Tf 3.539 0 Td [(x)]TJ/F156 6.6948 Tf 4.069 -1.434 Td [(\015)]TJ/F156 9.5641 Tf 4.931 1.434 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(j)]TJ/F156 9.5641 Tf 2.395 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(j)]TJ/F108 8.2889 Tf 5.392 0 Td [(j)]TJ/F158 9.5641 Tf 5.053 0 Td [(2)]TJ/F67 8.6077 Tf 8.672 0 Td [(Z)]TJ/F156 9.5641 Tf 5.739 0 Td [(;)-167(\015)]TJ/F158 9.5641 Tf 13.747 0 Td [(2)]TJ/F156 9.2055 Tf 8.673 0 Td [(\000)]TJ/F158 9.5641 Tf 7.567 0 Td [(g)]TJ/F156 9.5641 Tf 3.538 0 Td [(:)]TJ ET 1 0 0 1 40.867 488.085 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -488.085 cm BT /F21 8.2889 Tf 40.867 472.455 Td [(Here)]TJ/F156 9.2055 Tf 20.374 0 Td [(\000)]TJ/F21 8.2889 Tf 9.639 0 Td [(denotes)-250(the)-250(set)-250(of)-250(roots)-250(belonging)-250(to)]TJ/F80 8.6077 Tf 140.985 0 Td [(g)]TJ/F21 5.8022 Tf 4.335 -1.434 Td [(1)]TJ/F21 8.2889 Tf 3.858 1.434 Td [(.)]TJ ET 1 0 0 1 40.867 472.455 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -472.455 cm BT /F21 8.2889 Tf 52.822 462.029 Td [(For)-274(a)-275(standard)]TJ/F158 9.5641 Tf 57.669 0 Td [(Q)]TJ/F80 8.6077 Tf -0.576 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(-module)-274(of)-275(level)]TJ/F108 8.2889 Tf 64.992 0 Td [(k)]TJ/F158 9.5641 Tf 7.484 0 Td [(D)]TJ/F156 9.2055 Tf 10.634 0 Td [(\003)]TJ/F156 9.5641 Tf 7.327 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(c)]TJ/F156 9.5641 Tf 4.314 0 Td [(/)]TJ/F21 8.2889 Tf 5.841 0 Td [(with)-274(a)-275(highest)-274(weight)-274(vector)]TJ/F156 9.5641 Tf 112.145 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F21 8.2889 Tf 7.901 1.435 Td [(define)-274(Feigin\025Stoyanovsky's)-275(type)-274(subspace)]TJ ET 1 0 0 1 40.867 462.029 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -462.029 cm BT /F108 8.2889 Tf 40.867 451.568 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F21 8.2889 Tf 5.639 0 Td [(as)-250(the)-250(following)-250(subspace)-250(of)]TJ/F108 8.2889 Tf 109.28 0 Td [(L)]TJ/F156 9.5641 Tf 3.929 0 Td [(.)]TJ/F156 9.2055 Tf 3.568 0 Td [(\003)]TJ/F156 9.5641 Tf 7.327 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(:)]TJ ET 1 0 0 1 40.867 451.568 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -451.568 cm BT /F108 8.2889 Tf 64.777 435.938 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F156 9.2055 Tf 3.567 0 Td [(\003)]TJ/F156 9.5641 Tf 7.328 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F108 8.2889 Tf 10.117 0 Td [(U)]TJ/F156 9.5641 Tf 6.502 0 Td [(.)]TJ/F158 9.5641 Tf 4.143 0 Td [(Q)]TJ/F80 8.6077 Tf -0.575 0 Td [(g)]TJ/F21 5.8022 Tf 4.335 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.858 1.435 Td [(/)]TJ/F158 9.5641 Tf 5.693 0 Td [(\001)]TJ/F156 9.5641 Tf 4.784 0 Td [(v)]TJ/F156 6.4438 Tf 4.658 -1.435 Td [(\003)]TJ/F156 9.5641 Tf 5.627 1.435 Td [(:)]TJ ET 1 0 0 1 40.867 435.938 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -435.938 cm BT /F21 8.2889 Tf 52.822 420.342 Td [(By)-202(using)-202(a)-203(description)-202(of)-202(combinatorial)-202(bases)-203(for)-202(Feigin\025Stoyanovsky's)-202(type)-202(subspaces)-203(for)-202(affine)-202(Lie)-202(algebra)]TJ/F80 8.6077 Tf 403.938 0 Td [(sl)]TJ/F156 9.5641 Tf 6.217 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(l)]TJ/F158 9.5641 Tf 3.179 0 Td [(C)]TJ/F21 8.2889 Tf 8.368 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F67 8.6077 Tf 4.76 0 Td [(C)]TJ/F156 9.5641 Tf 6.216 0 Td [(/)]TJ/F159 6.6948 Tf 1.084 3.472 Td [(e)]TJ/F21 8.2889 Tf 4.658 -3.472 Td [(in)]TJ ET 1 0 0 1 40.867 420.342 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -420.342 cm BT /F21 8.2889 Tf 40.867 409.881 Td [(terms)-260(of)]TJ/F156 9.5641 Tf 34.001 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F156 9.5641 Tf 4.31 0 Td [(;)-167(`)]TJ/F158 9.5641 Tf 11.447 0 Td [(C)]TJ/F21 8.2889 Tf 9.67 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(-admissible)-260(vectors,)-260(as)-260(well)-260(as)-260(operators)]TJ/F156 9.5641 Tf 155.914 0 Td [(')]TJ/F21 5.8022 Tf 5.375 -1.434 Td [(0)]TJ/F156 9.5641 Tf 3.858 1.434 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(')]TJ/F108 5.8022 Tf 27.566 -1.434 Td [(m)]TJ/F158 6.6948 Tf 5.013 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 6.013 1.434 Td [(\050constructed)-260(with)-260(certain)-260(intertwining)-260(operators)]TJ ET 1 0 0 1 40.867 409.881 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -409.881 cm BT /F21 8.2889 Tf 40.867 399.42 Td [(between)-247(standard)-247(modules\051)-247(and)-247(a)-247(simple)-247(current)-247(operator)]TJ/F158 9.5641 Tf 224.375 0 Td [(T)]TJ/F156 9.5641 Tf 3.184 0 Td [(!)]TJ/F158 9.5641 Tf 6.456 0 Td [(U)]TJ/F21 8.2889 Tf 3.185 0 Td [(,)-247(we)-247(obtain)-247(exact)-247(sequences)-247(of)-247(these)-247(subspaces)-247(at)-247(fixed)-248(level)]TJ ET 1 0 0 1 40.867 399.42 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -399.42 cm BT /F108 8.2889 Tf 40.867 388.96 Td [(k)]TJ/F21 8.2889 Tf 6.382 0 Td [(\050cf.)]TJ ET 1 0 0 1 61.066 388.96 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -61.066 -388.96 cm BT /F21 8.2889 Tf 61.066 388.96 Td [(Theorem)-250(5.1)]TJ ET 1 0 0 1 108.835 388.96 cm 0 g 0 G 1 0 0 1 -108.835 -388.96 cm BT /F21 8.2889 Tf 108.835 388.96 Td [(\051:)]TJ ET 1 0 0 1 40.867 388.96 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -388.96 cm BT /F21 8.2889 Tf 64.777 367.694 Td [(0)]TJ/F158 9.5641 Tf 7.456 0 Td [(!)]TJ/F108 8.2889 Tf 13.167 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.018 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.685 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 6.6948 Tf 8.578 8.87 Td [(T)]TJ/F156 6.6948 Tf 2.23 0 Td [(!)]TJ/F158 6.6948 Tf 4.519 0 Td [(U)]TJ/F158 5.7385 Tf 2.229 2.43 Td [(\012)]TJ/F108 4.9734 Tf 4.591 0 Td [(k)]TJ/F158 9.5641 Tf -15.614 -8.473 Td [(\000)466(\000)222(\000)465(!)]TJ/F108 8.2889 Tf 24.516 0 Td [(W)]TJ/F156 6.6948 Tf 13.949 6.396 Td [(')]TJ/F21 4.9734 Tf 3.762 -1.353 Td [(0)]TJ/F158 9.5641 Tf -5.808 -5.043 Td [(\000)588(!)]TJ/F159 9.5641 Tf 21.258 9.086 Td [(X)]TJ/F108 5.8022 Tf -6.254 -19.69 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F158 6.6948 Tf 3.378 1.353 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F21 4.9734 Tf 4.027 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.377 1.353 Td [(.)]TJ/F108 5.8022 Tf 2.498 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F108 8.2889 Tf 4.091 10.604 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F21 4.9734 Tf 1.747 -1.353 Td [(1)]TJ/F156 6.6948 Tf 8.578 9.14 Td [(')]TJ/F21 4.9734 Tf 3.762 -1.353 Td [(1)]TJ/F158 9.5641 Tf -5.808 -4.999 Td [(\000)588(!)-278(\001)-166(\001)-167(\001)]TJ/F156 6.6948 Tf 30.871 6.828 Td [(')]TJ/F108 4.9734 Tf 3.762 -1.353 Td [(m)]TJ/F158 5.7385 Tf 4.297 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf -14.581 -5.475 Td [(\000)225(\000)226(!)]TJ/F108 8.2889 Tf 23.777 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.747 -1.004 Td [(m)]TJ/F158 9.5641 Tf 7.95 2.439 Td [(!)]TJ/F21 8.2889 Tf 13.167 0 Td [(0)]TJ/F156 9.5641 Tf 4.799 0 Td [(:)]TJ ET 1 0 0 1 40.867 367.694 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -367.694 cm BT /F21 8.2889 Tf 40.867 341.939 Td [(Here)]TJ/F108 8.2889 Tf 20.626 0 Td [(W)]TJ/F158 9.5641 Tf 12.549 0 Td [(D)]TJ/F108 8.2889 Tf 10.764 0 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F21 5.8022 Tf 4.311 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.857 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.328 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.241 1.435 Td [(C)-249(\001)-167(\001)-167(\001)-249(C)]TJ/F108 8.2889 Tf 33.237 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.631 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/)]TJ/F21 8.2889 Tf 5.892 0 Td [(is)-280(a)-281(Feigin\025Stoyanovsky's)-280(type)-281(subspace)-280(for)-280(level)]TJ/F108 8.2889 Tf 189.035 0 Td [(k)]TJ/F158 9.5641 Tf 7.614 0 Td [(D)]TJ/F108 8.2889 Tf 10.763 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.242 1.435 Td [(C)-249(\001)-167(\001)-167(\001)-249(C)]TJ/F108 8.2889 Tf 33.236 0 Td [(k)]TJ/F156 6.6948 Tf 4.311 -1.435 Td [(`)]TJ/F21 8.2889 Tf 3.631 1.435 Td [(,)]TJ/F108 8.2889 Tf 4.264 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.018 -1.393 Td [(`)]TJ/F156 6.6948 Tf 3.183 1.393 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(1)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F158 9.5641 Tf 7.179 2.828 Td [(D)]TJ ET 1 0 0 1 40.867 341.939 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -341.939 cm BT /F108 8.2889 Tf 40.867 331.478 Td [(W)]TJ/F156 9.5641 Tf 9.246 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(k)]TJ/F156 6.6948 Tf 4.31 -1.435 Td [(`)]TJ/F156 9.2055 Tf 3.632 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(0)]TJ/F158 9.5641 Tf 6.05 1.435 Td [(C)]TJ/F108 8.2889 Tf 9.652 0 Td [(k)]TJ/F21 5.8022 Tf 4.31 -1.435 Td [(0)]TJ/F156 9.2055 Tf 3.858 1.435 Td [(\003)]TJ/F21 5.8022 Tf 7.327 -1.435 Td [(1)]TJ/F158 9.5641 Tf 6.05 1.435 Td [(C)-229(\001)-167(\001)-167(\001)-229(C)]TJ/F108 8.2889 Tf 32.66 0 Td [(k)]TJ/F156 6.6948 Tf 4.311 -1.435 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(\000)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F156 9.2055 Tf 3.857 1.435 Td [(\003)]TJ/F156 6.6948 Tf 7.328 -1.435 Td [(`)]TJ/F156 9.5641 Tf 3.631 1.435 Td [(/)]TJ/F21 8.2889 Tf 3.568 0 Td [(,)-258(and)]TJ/F108 8.2889 Tf 20.462 0 Td [(W)]TJ/F108 5.8022 Tf 8.132 -1.435 Td [(I)]TJ/F21 4.9734 Tf 1.746 -1.353 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.788 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F108 4.9734 Tf 1.746 -1.004 Td [(m)]TJ/F21 8.2889 Tf 7.431 2.439 Td [(Feigin\025Stoyanovsky's)-258(type)-258(subspaces)-257(derived)-258(from)]TJ/F108 8.2889 Tf 195.146 0 Td [(W)]TJ/F21 8.2889 Tf 11.383 0 Td [(by)-258(a)-258(procedure)]TJ ET 1 0 0 1 40.867 331.478 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -331.478 cm BT /F21 8.2889 Tf 40.867 321.017 Td [(described)-250(in)-250(Section)]TJ ET 1 0 0 1 119.519 321.017 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -119.519 -321.017 cm BT /F21 8.2889 Tf 119.519 321.017 Td [(5.1)]TJ ET 1 0 0 1 131.066 321.017 cm 0 g 0 G 1 0 0 1 -131.066 -321.017 cm BT /F21 8.2889 Tf 131.066 321.017 Td [(.)]TJ ET 1 0 0 1 40.867 321.017 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -321.017 cm BT /F21 8.2889 Tf 52.822 310.59 Td [(Exact)-304(sequences)-305(described)-304(above)-305(directly)-304(lead)-305(to)-304(systems)-304(of)-305(relations)-304(among)-305(formal)-304(characters)-305(of)-304(those)-304(subspaces)-305(\050cf.)]TJ ET 1 0 0 1 40.867 310.59 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -310.59 cm BT /F21 8.2889 Tf 40.867 300.13 Td [(Eq.)]TJ ET 1 0 0 1 54.419 300.13 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -54.419 -300.13 cm BT /F21 8.2889 Tf 54.419 300.13 Td [(\0506.2\051)]TJ ET 1 0 0 1 72.091 300.13 cm 0 g 0 G 1 0 0 1 -72.091 -300.13 cm BT /F21 8.2889 Tf 72.091 300.13 Td [(\051:)]TJ ET 1 0 0 1 40.867 300.13 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -300.13 cm BT /F159 9.5641 Tf 67.89 292.216 Td [(X)]TJ/F108 5.8022 Tf -3.113 -19.69 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(2)]TJ/F108 5.8022 Tf 4.211 0 Td [(D)]TJ/F156 6.6948 Tf 4.027 0 Td [(.)]TJ/F108 5.8022 Tf 2.497 0 Td [(K)]TJ/F156 6.6948 Tf 4.308 0 Td [(/)]TJ/F156 9.5641 Tf 2.497 10.604 Td [(.)]TJ/F158 9.5641 Tf 3.568 0 Td [(\000)]TJ/F21 8.2889 Tf 7.46 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(/)]TJ/F158 6.6948 Tf 3.567 3.95 Td [(j)]TJ/F108 5.8022 Tf 1.915 0 Td [(I)]TJ/F158 6.6948 Tf 2.219 0 Td [(j)]TJ/F156 9.5641 Tf 2.413 -3.95 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(I)]TJ/F156 9.5641 Tf 2.718 1.435 Td [(/.)]TJ/F108 8.2889 Tf 7.134 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F156 9.5641 Tf 3.857 1.435 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.192 0 Td [(z)]TJ/F156 6.6948 Tf 3.713 -1.435 Td [(`)]TJ/F158 9.5641 Tf 3.632 1.435 Td [(I)]TJ/F108 8.2889 Tf 5.008 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(D)]TJ/F156 9.5641 Tf 10.117 0 Td [(.)]TJ/F108 8.2889 Tf 3.567 0 Td [(z)]TJ/F21 5.8022 Tf 3.714 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.857 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 9.5641 Tf 5.47 -2.597 Td [(:)-167(:)-166(:)-167(.)]TJ/F108 8.2889 Tf 16.24 0 Td [(z)]TJ/F156 6.6948 Tf 3.713 -1.435 Td [(`)]TJ/F108 8.2889 Tf 3.631 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/)]TJ/F108 5.8022 Tf 3.568 3.95 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.393 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.875 -2.557 Td [(\037)-100(.)]TJ/F108 8.2889 Tf 10.148 0 Td [(W)]TJ/F108 5.8022 Tf 8.131 -1.435 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F156 6.6948 Tf 3.184 1.392 Td [(;)]TJ/F108 5.8022 Tf 2.216 0 Td [(k)]TJ/F21 4.9734 Tf 3.017 -1.353 Td [(0)]TJ/F156 6.6948 Tf 3.378 1.353 Td [(;:::;)]TJ/F108 5.8022 Tf 9.955 0 Td [(k)]TJ/F156 5.7385 Tf 3.017 -1.392 Td [(`)]TJ/F158 5.7385 Tf 2.686 0 Td [(\000)]TJ/F21 4.9734 Tf 4.476 0 Td [(1)]TJ/F156 9.5641 Tf 3.876 2.827 Td [(/.)]TJ/F108 8.2889 Tf 7.135 0 Td [(z)]TJ/F21 5.8022 Tf 3.713 -1.435 Td [(1)]TJ/F108 8.2889 Tf 3.858 1.435 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(;)-167(:)-166(:)-167(:)-167(;)]TJ/F108 8.2889 Tf 22.191 0 Td [(z)]TJ/F156 6.6948 Tf 3.714 -1.435 Td [(`)]TJ/F108 8.2889 Tf 3.631 1.435 Td [(q)]TJ/F158 9.5641 Tf 4.451 0 Td [(I)]TJ/F108 8.2889 Tf 5.009 0 Td [(q)]TJ/F156 9.5641 Tf 4.451 0 Td [(/:)]TJ ET 1 0 0 1 40.867 283.13 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -283.13 cm BT /F21 8.2889 Tf 52.822 257.806 Td [(The)-206(paper)-206(is)-206(organized)-205(as)-206(follows.)-206(Section)]TJ ET 1 0 0 1 210.611 257.806 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -210.611 -257.806 cm BT /F21 8.2889 Tf 210.611 257.806 Td [(2)]TJ ET 1 0 0 1 215.41 257.806 cm 0 g 0 G 1 0 0 1 -215.41 -257.806 cm BT /F21 8.2889 Tf 217.117 257.806 Td [(gives)-206(the)-206(setting.)-206(In)-206(Section)]TJ ET 1 0 0 1 321.429 257.806 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -321.429 -257.806 cm BT /F21 8.2889 Tf 321.429 257.806 Td [(3)]TJ ET 1 0 0 1 326.228 257.806 cm 0 g 0 G 1 0 0 1 -326.228 -257.806 cm BT /F21 8.2889 Tf 327.934 257.806 Td [(we)-206(define)-206(Feigin\025Stoyanovsky's)-206(type)-205(subspaces)]TJ ET 1 0 0 1 40.867 257.806 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -257.806 cm BT /F21 8.2889 Tf 40.867 247.345 Td [(and)-287(present)-286(the)-287(result)-287(on)-287(combinatorial)-286(bases.)-287(Section)]TJ ET 1 0 0 1 251.414 247.345 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -251.414 -247.345 cm BT /F21 8.2889 Tf 251.414 247.345 Td [(4)]TJ ET 1 0 0 1 256.214 247.345 cm 0 g 0 G 1 0 0 1 -256.214 -247.345 cm BT /F21 8.2889 Tf 258.591 247.345 Td [(gives)-287(the)-286(vertex)-287(operator)-287(construction)-287(for)-286(fundamental)-287(modules,)]TJ ET 1 0 0 1 40.867 247.345 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -247.345 cm BT /F21 8.2889 Tf 40.867 236.885 Td [(and)-266(introduces)-266(the)-266(intertwining)-267(operators)-266(and)-266(a)-266(simple)-266(current)-267(operato)1(r.)-267(The)-266(last)-266(two)-266(sections)-266(contain)-267(the)-266(main)-266(results)-266(of)]TJ ET 1 0 0 1 40.867 236.885 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -236.885 cm BT /F21 8.2889 Tf 40.867 226.424 Td [(the)-228(paper:)-228(Section)]TJ ET 1 0 0 1 111.888 226.424 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -111.888 -226.424 cm BT /F21 8.2889 Tf 111.888 226.424 Td [(5)]TJ ET 1 0 0 1 116.687 226.424 cm 0 g 0 G 1 0 0 1 -116.687 -226.424 cm BT /F21 8.2889 Tf 118.577 226.424 Td [(states)-228(the)-228(result)-228(on)-228(exactness)-228(\050cf.)]TJ ET 1 0 0 1 245.193 226.424 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -245.193 -226.424 cm BT /F21 8.2889 Tf 245.193 226.424 Td [(Theorem)-228(5.1)]TJ ET 1 0 0 1 292.78 226.424 cm 0 g 0 G 1 0 0 1 -292.78 -226.424 cm BT /F21 8.2889 Tf 292.78 226.424 Td [(\051,)-228(while)-228(in)-228(Section)]TJ ET 1 0 0 1 362.309 226.424 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -362.309 -226.424 cm BT /F21 8.2889 Tf 362.309 226.424 Td [(6)]TJ ET 1 0 0 1 367.109 226.424 cm 0 g 0 G 1 0 0 1 -367.109 -226.424 cm BT /F21 8.2889 Tf 368.999 226.424 Td [(we)-228(obtain)-228(the)-228(corresponding)-228(system)]TJ ET 1 0 0 1 40.867 226.424 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -226.424 cm BT /F21 8.2889 Tf 40.867 215.963 Td [(of)-223(relations)-224(among)-223(characters)-224(of)-223(Feigin\025Stoyanovsky's)-224(type)-223(subspaces)-223(at)-224(fixed)-223(level)-224(\050cf.)-223(Eqs.)]TJ ET 1 0 0 1 387.657 215.963 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -387.657 -215.963 cm BT /F21 8.2889 Tf 387.657 215.963 Td [(\0506.2\051)]TJ ET 1 0 0 1 405.329 215.963 cm 0 g 0 G 1 0 0 1 -405.329 -215.963 cm BT /F21 8.2889 Tf 407.181 215.963 Td [(and)]TJ ET 1 0 0 1 423.282 215.963 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -423.282 -215.963 cm BT /F21 8.2889 Tf 423.282 215.963 Td [(\0506.4\051)]TJ ET 1 0 0 1 440.953 215.963 cm 0 g 0 G 1 0 0 1 -440.953 -215.963 cm BT /F21 8.2889 Tf 440.953 215.963 Td [(\051,)-224(and)-223(present)-223(the)]TJ ET 1 0 0 1 40.867 215.963 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -215.963 cm BT /F21 8.2889 Tf 40.867 205.502 Td [(proof)-250(that)-250(such)-250(system)-250(has)-250(a)-250(unique)-250(solution.)]TJ ET 1 0 0 1 40.867 205.502 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -205.502 cm BT /F21 8.2889 Tf 52.822 195.076 Td [(I)-250(sincerely)-250(thank)-250(Mirko)-250(Primc)-250(for)-250(his)-250(valuable)-250(suggestions.)]TJ ET 1 0 0 1 40.867 195.076 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -195.076 cm BT /F123 8.2889 Tf 40.867 172.352 Td [(2.)-500(Affine)-250(Lie)-250(algebra)]TJ/F180 8.6077 Tf 83.128 0 Td [(sl)]TJ/F190 9.5641 Tf 7.398 0 Td [(.)]TJ/F174 8.2889 Tf 3.616 0 Td [(l)]TJ/F191 9.5641 Tf 5.254 0 Td [(C)]TJ/F123 8.2889 Tf 9.585 0 Td [(1)]TJ/F190 9.5641 Tf 4.824 0 Td [(;)]TJ/F67 8.6077 Tf 5.363 0 Td [(C)]TJ/F190 9.5641 Tf 6.216 0 Td [(/)]TJ/F192 6.6948 Tf 1.729 3.471 Td [(e)]TJ/F123 8.2889 Tf 4.438 -3.471 Td [(and)-250(standard)-250(modules)]TJ ET 1 0 0 1 40.867 172.352 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -172.352 cm BT /F21 8.2889 Tf 52.822 151.601 Td [(Let)]TJ/F80 8.6077 Tf 13.633 0 Td [(g)]TJ/F158 9.5641 Tf 6.992 0 Td [(D)]TJ/F80 8.6077 Tf 10.117 0 Td [(sl)]TJ/F156 9.5641 Tf 6.217 0 Td [(.`)]TJ/F158 9.5641 Tf 10.086 0 Td [(C)]TJ/F21 8.2889 Tf 9.503 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)]TJ/F67 8.6077 Tf 4.759 0 Td [(C)]TJ/F156 9.5641 Tf 6.217 0 Td [(/)]TJ/F21 8.2889 Tf 3.567 0 Td [(,)]TJ/F156 9.5641 Tf 3.986 0 Td [(`)]TJ/F158 9.5641 Tf 7.132 0 Td [(2)]TJ/F67 8.6077 Tf 8.673 0 Td [(N)]TJ/F21 8.2889 Tf 6.216 0 Td [(,)-247(and)]TJ/F80 8.6077 Tf 20.28 0 Td [(h)]TJ/F21 8.2889 Tf 6.532 0 Td [(a)-247(Cartan)-247(subalgebra)-246(of)]TJ/F80 8.6077 Tf 86.619 0 Td [(g)]TJ/F21 8.2889 Tf 4.336 0 Td [(.)-247(Denote)-247(by)]TJ/F108 8.2889 Tf 44.904 0 Td [(R)]TJ/F21 8.2889 Tf 6.911 0 Td [(the)-247(corresponding)-247(root)-246(system)-247(\050identified)-247(in)-247(the)]TJ ET 1 0 0 1 40.867 151.601 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -151.601 cm BT /F21 8.2889 Tf 40.867 141.14 Td [(usual)-250(way)-250(as)-250(a)-250(subset)-250(of)]TJ/F67 8.6077 Tf 93.191 0 Td [(R)]TJ/F156 6.6948 Tf 6.217 3.472 Td [(`)]TJ/F158 6.6948 Tf 3.133 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 3.857 -3.472 Td [(\051:)]TJ ET 1 0 0 1 40.867 141.14 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -141.14 cm BT /F108 8.2889 Tf 64.777 125.51 Td [(R)]TJ/F158 9.5641 Tf 7.522 0 Td [(D)-278(f\006)]TJ/F156 9.5641 Tf 21.115 0 Td [(.\017)]TJ/F108 5.8022 Tf 7.747 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.283 1.435 Td [(\000)]TJ/F156 9.5641 Tf 9.586 0 Td [(\017)]TJ/F108 5.8022 Tf 4.179 -1.435 Td [(j)]TJ/F156 9.5641 Tf 2.175 1.435 Td [(/)]TJ/F158 9.5641 Tf 6.224 0 Td [(j)]TJ/F21 8.2889 Tf 5.392 0 Td [(1)]TJ/F158 9.5641 Tf 7.456 0 Td [(\024)]TJ/F108 8.2889 Tf 9.868 0 Td [(i)]TJ/F156 9.5641 Tf 5.027 0 Td [(<)]TJ/F108 8.2889 Tf 10.117 0 Td [(j)]TJ/F158 9.5641 Tf 5.052 0 Td [(\024)]TJ/F156 9.5641 Tf 9.868 0 Td [(`)]TJ/F158 9.5641 Tf 6.601 0 Td [(C)]TJ/F21 8.2889 Tf 9.586 0 Td [(1)]TJ/F158 9.5641 Tf 4.799 0 Td [(g)]TJ/F156 9.5641 Tf 3.539 0 Td [(:)]TJ ET 1 0 0 1 40.867 125.51 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -125.51 cm BT /F21 8.2889 Tf 40.867 109.88 Td [(As)-302(usual,)-302(we)-302(fix)-302(simple)-302(roots)]TJ/F156 9.5641 Tf 113.132 0 Td [(\013)]TJ/F108 5.8022 Tf 5.461 -1.435 Td [(i)]TJ/F158 9.5641 Tf 5.918 1.435 Td [(D)]TJ/F156 9.5641 Tf 11.221 0 Td [(\017)]TJ/F108 5.8022 Tf 4.179 -1.435 Td [(i)]TJ/F158 9.5641 Tf 4.725 1.435 Td [(\000)]TJ/F156 9.5641 Tf 10.027 0 Td [(\017)]TJ/F108 5.8022 Tf 4.179 -1.435 Td [(i)]TJ/F158 6.6948 Tf 1.66 0 Td [(C)]TJ/F21 5.8022 Tf 5.222 0 Td [(1)]TJ/F21 8.2889 Tf 3.857 1.435 Td [(,)]TJ/F108 8.2889 Tf 4.443 0 Td [(i)]TJ/F158 9.5641 Tf 6.131 0 Td [(D)]TJ/F21 8.2889 Tf 11.221 0 Td [(1)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(`)]TJ/F21 8.2889 Tf 26.668 0 Td [(,)-302(and)-302(have)-302(triangular)-302(decomposition)]TJ/F80 8.6077 Tf 141.596 0 Td [(g)]TJ/F158 9.5641 Tf 8.097 0 Td [(D)]TJ/F80 8.6077 Tf 11.22 0 Td [(n)]TJ/F158 6.6948 Tf 4.534 -1.435 Td [(\000)]TJ/F158 9.5641 Tf 8.287 1.435 Td [(\010)]TJ/F80 8.6077 Tf 10.218 0 Td [(h)]TJ/F158 9.5641 Tf 7.054 0 Td [(\010)]TJ/F80 8.6077 Tf 10.218 0 Td [(n)]TJ/F158 6.6948 Tf 4.533 -1.435 Td [(C)]TJ/F21 8.2889 Tf 5.72 1.435 Td [(.)-302(Denote)]TJ ET 1 0 0 1 40.867 109.88 cm 0 g 0 G 0 g 0 G 1 0 0 1 -40.867 -109.88 cm BT /F21 8.2889 Tf 40.867 99.419 Td [(by)]TJ/F108 8.2889 Tf 12.261 0 Td [(Q)]TJ/F158 9.5641 Tf 11.764 0 Td [(D)]TJ/F108 8.2889 Tf 12.291 0 Td [(Q)]TJ/F156 9.5641 Tf 6.933 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(R)]TJ/F156 9.5641 Tf 4.865 0 Td [(/)]TJ/F21 8.2889 Tf 6.488 0 Td [(the)-352(root)-353(lattice)-352(and)-352(by)]TJ/F108 8.2889 Tf 90.327 0 Td [(P)]TJ/F158 9.5641 Tf 10.073 0 Td [(D)]TJ/F108 8.2889 Tf 12.29 0 Td [(P)]TJ/F156 9.5641 Tf 5.243 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(R)]TJ/F156 9.5641 Tf 4.865 0 Td [(/)]TJ/F21 8.2889 Tf 6.488 0 Td [(the)-352(weight)-352(lattice)-353(of)]TJ/F108 8.2889 Tf 81.961 0 Td [(R)]TJ/F21 8.2889 Tf 4.865 0 Td [(.)-353(Let)]TJ/F156 9.5641 Tf 19.381 0 Td [(!)]TJ/F108 5.8022 Tf 6.265 -1.434 Td [(i)]TJ/F21 8.2889 Tf 2.157 1.434 Td [(,)]TJ/F108 8.2889 Tf 4.861 0 Td [(i)]TJ/F158 9.5641 Tf 7.201 0 Td [(D)]TJ/F21 8.2889 Tf 12.29 0 Td [(1)]TJ/F156 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[(])-286(gave)-285(another)-286(proof)-286(and)-285(combinatorial)]TJ ET 1 0 0 1 37.134 290.906 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -290.906 cm BT /F21 8.2889 Tf 37.134 280.445 Td [(interpretation)-466(of)-466(these)-466(character)-465(formulas,)-467(one)-465(of)-466(several)-466(new)-466(ingredients)-466(in)-466(the)-466(proof)-466(being)-465(the)-466(so-called)-466(principal)]TJ ET 1 0 0 1 37.134 280.445 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -280.445 cm BT /F21 8.2889 Tf 37.134 269.984 Td [(subspaces,)-321(later)-320(also)-321(named)-320(Feigin\025Stoyanovsky's)-320(subspaces.)-321(By)-321(using)-320(intertwining)-321(operators)-320(Georgiev)-321(\050cf.)-321([)]TJ ET 1 0 0 1 454.22 269.984 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -454.22 -269.984 cm BT /F21 8.2889 Tf 454.22 269.984 Td [(5)]TJ ET 1 0 0 1 459.019 269.984 cm 0 g 0 G 1 0 0 1 -459.019 -269.984 cm BT /F21 8.2889 Tf 459.019 269.984 Td [(]\051)-321(extended)]TJ ET 1 0 0 1 37.134 269.984 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -269.984 cm BT 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9.5641 Tf 6.217 0 Td [(/)]TJ/F159 6.6948 Tf 1.084 3.472 Td [(e)]TJ/F21 8.2889 Tf 2.982 -3.472 Td [(-modules.)-177(It)-176(turned)-176(out)-176(that)-176(these)-176(recurrence)-177(relations)-176(are)-176(precisely)-176(the)]TJ ET 1 0 0 1 37.134 228.141 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -228.141 cm BT /F21 8.2889 Tf 37.134 217.68 Td [(known)-261(recursion)-261(formulas)-261(of)-260(Rogers)-261(and)-261(Selberg,)-261(already)-261(solved)-261(by)-260(G.)-261(Andrews)-261(while)-261(working)-261(on)-261(Gordon)-261(identities)-260(\050cf.)-261([)]TJ ET 1 0 0 1 496.647 217.68 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -496.647 -217.68 cm BT /F21 8.2889 Tf 496.647 217.68 Td [(8)]TJ ET 1 0 0 1 501.446 217.68 cm 0 g 0 G 1 0 0 1 -501.446 -217.68 cm BT /F21 8.2889 Tf 501.446 217.68 Td [(,)]TJ ET 1 0 0 1 37.134 217.68 cm 0 g 0 G 0 g 0 G 1 0 0 1 0 -10.461 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -37.134 -207.219 cm BT /F21 8.2889 Tf 37.134 207.219 Td [(9)]TJ ET 1 0 0 1 41.933 207.219 cm 0 g 0 G 1 0 0 1 -41.933 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2.982 -3.471 Td [(-modules)-233(and)-234(thus)-233(acquire)-234(a)-233(recurrence)-233(system)-234(for)-233(characters)-234(of)-233(these)]TJ ET 1 0 0 1 37.134 196.759 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -196.759 cm BT /F21 8.2889 Tf 37.134 186.298 Td [(subspaces.)-257(Furthermore,)-257(in)-257([)]TJ ET 1 0 0 1 144.98 186.298 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -144.98 -186.298 cm BT /F21 8.2889 Tf 144.98 186.298 Td [(11)]TJ ET 1 0 0 1 154.578 186.298 cm 0 g 0 G 1 0 0 1 -154.578 -186.298 cm BT /F21 8.2889 Tf 154.578 186.298 Td [(])-257(Calinescu)-257(used)-257(the)-257(exact)-257(sequence)-257(method)-257(to)-257(obtain)-257(systems)-257(of)-256(recurrences)-257(for)-257(characters)]TJ ET 1 0 0 1 37.134 186.298 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -186.298 cm BT /F21 8.2889 Tf 37.134 174.587 Td [(of)-316(all)-317(principal)-316(subspaces)-316(of)-317(arbitrary)-316(level)-316(standard)]TJ/F80 8.6077 Tf 203.183 0 Td [(sl)]TJ/F156 9.5641 Tf 6.217 0 Td [(.)]TJ/F21 8.2889 Tf 3.567 0 Td [(3)]TJ/F156 9.5641 Tf 4.799 0 Td [(;)]TJ/F67 8.6077 Tf 4.76 0 Td [(C)]TJ/F156 9.5641 Tf 6.217 0 Td [(/)]TJ/F159 6.6948 Tf 1.083 3.472 Td [(e)]TJ/F21 8.2889 Tf 2.982 -3.472 Td [(-modules)-316(and)-317(the)-316(corresponding)-316(characters)-317(for)-316(some)-316(classes)]TJ ET 1 0 0 1 37.134 174.587 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -174.587 cm BT /F21 8.2889 Tf 37.134 164.126 Td [(of)-276(principal)-275(subspaces.)-276(Finally,)-275(this)-276(approach)-275(was)-276(further)-275(developed)-276(in)-276([)]TJ ET 1 0 0 1 311.434 164.126 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -311.434 -164.126 cm BT /F21 8.2889 Tf 311.434 164.126 Td [(12)]TJ ET 1 0 0 1 321.033 164.126 cm 0 g 0 G 1 0 0 1 -321.033 -164.126 cm BT /F21 8.2889 Tf 321.033 164.126 Td [(,)]TJ ET 1 0 0 1 322.972 164.126 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -322.972 -164.126 cm BT /F21 8.2889 Tf 322.972 164.126 Td [(13)]TJ ET 1 0 0 1 332.571 164.126 cm 0 g 0 G 1 0 0 1 -332.571 -164.126 cm BT /F21 8.2889 Tf 332.571 164.126 Td [(])-276(where)-275(presentations)-276(of)-275(principal)-276(subspaces)]TJ ET 1 0 0 1 37.134 164.126 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -164.126 cm BT /F21 8.2889 Tf 37.134 153.665 Td [(of)-250(all)-250(standard)]TJ/F80 8.6077 Tf 56.588 0 Td [(sl)]TJ/F156 9.5641 Tf 6.216 0 Td [(.)]TJ/F108 8.2889 Tf 3.568 0 Td [(l)]TJ/F158 9.5641 Tf 4.396 0 Td [(C)]TJ/F21 8.2889 Tf 9.585 0 Td [(1)]TJ/F156 9.5641 Tf 4.8 0 Td [(;)]TJ/F67 8.6077 Tf 4.759 0 Td [(C)]TJ/F156 9.5641 Tf 6.217 0 Td [(/)]TJ/F159 6.6948 Tf 1.084 3.472 Td [(e)]TJ/F21 8.2889 Tf 2.982 -3.472 Td [(-modules)-250(are)-250(given.)]TJ ET 1 0 0 1 37.134 153.665 cm 0 g 0 G 0 g 0 G 1 0 0 1 -37.134 -153.665 cm BT /F21 8.2889 Tf 49.089 142.707 Td [(Another)-376(construction)-377(of)-376(combinatorial)-377(bases)-376(of)-377(standard)-376(modules)-377(of)-376(affine)-377(Lie)-376(algebras)-377(was)-376(given)-377(in)-376([)]TJ ET 1 0 0 1 449.17 142.707 cm 0 0 .4 rg 0 0 .4 RG 1 0 0 1 -449.17 -142.707 cm BT /F21 8.2889 Tf 449.17 142.707 Td 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