CONSTRUCTION DE FONCTEURS ENTRE CATEGORIES

Transcription

CONSTRUCTION DE FONCTEURS ENTRE CATEGORIES
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ȎgY=A Yhc"Z[Wm U ( ¼  …de_s](['[r_smec¤Žd±‹ meg^žWYfklc†m‡_^g n0Y S4VU 7nb…2g3[ S4 ¼ zUȎdedeYY2[Wc}cJYhded‡Y®¯`bY[Wm
u, S4 ¼ w…2de_^]“[
¤ { 9Q
* S4VU "Us¤†³´WuXi S46u6U
l}_sg3k
¤ { …
`bg abde`‘[-\^](…2g3nšY"p dhYhp iY"g0cRu6U[rm!S B
( u3UbwYhc'¤ { Y2[rcŠ›wmonwY
[Wmegb_sgz}d‡_^]([
4 ¼ S
4 h
S VU
"
"
¸
u4 ¼ Bh
"
"
u4 ¼
l}_sg3k
\4 ¼
S
"
"
u4 ¼ "
"
u4 ¼ l
‹ _ `>­ d‡YdeY"iiYs^ab`bmy[†®|`bY[rmu* GYf[rcŠc†Y"d7®¯`bY u 3Uc[S3b…d‡_^]“[`u, \S4 ¼ ±z
,-.qqµ. t”14 E 57‚#ƒh435„ƒE&(13/24u!5 @.97.¶J$ :Ž/2ˆ43/ƒ
1:,q. 7/24 „!
5 gg M2Bg A N g .0ƒ ‚ @5 2qIƒ - ‚ ‡.0=ƒ Nrbƒ57‚#ƒh4357ƒl&(13/2435@@.‚„43/Oqr2
7.J
$ :
8P:9
. !2ˆ5„ƒ gg M24
g A N g gg M2N
g A N g M2g gg :
ȎgY=A Yhc"žE‹ …m
A
Ng ³kh  ´ ²³r$zS>$´k.
%
! 0 % !
YhcŽd‡Y-ab]J_wn0`bm‡c gg M³W$vS$´j. Y2[WcŽg0`Ud±s[†…`w’[Wm ? ³ ´R $zOR'] ? ³ ´R &g„³v´“|n0_sg3k
Y2[rckE_^ia3_|[wYhp YnwY[r_^`‘[·\s]†_s`ba3Y2[gb_^]Ji…`b˜ n0Y!$ZY"cYhg>a3…2]†c†mykl`bd‡meYh]' •meg0’ $
nZ‹ _ ` ­ do…ab]†Y"imYh­ ]JY'…^[J[rYh]Jc†me_sgz
}d‡_^]([dy…[rYfkl_^g3nwY…s[†[rY"]†c†m‡_^g£Yf[rcc†]Jme›wmo…d‡YhiYhgwcŠ›w](…m‡Y[Wmq g‹ Y2[rca3…s[`bgµ[W_^`3[W€¦\s]†_s`ba3Y
gb_s]†i…2dn0Y$zZȎc'[Wm7 $

gg M2N
g A Ng ³kh  ´ gg M2g³r$VS>$´j. %
! ! )$
Ž…2]ŠdeYiYhm iY'…2]†\s`biYhgwchwab`bmo[J®¯`UY ³ ´Rš[r`ba & cg7³^´(2žE‹²…2m7…`3[J[rm
g g
A N M2g g ³jh  ´ ³W$VS$´j. M2g gg
%
! +! )$
!
$
l ‹ …`bcJ]†Ya3…]Jchbd‹†Yhp d"Yhp iYhg|c gg Y2[rc'kE_^ijbmeg3…2mo[W_^gd‡megY2p …2me]JYn0Yc†Yh]JiY2[@n0Ydo…
’ˆ_^]JiY
³r$zS>$´j. ³¦ –´(3_ `­ a3…]“kl_^`b]Jc}d‡Y2[-[W_^`3[W€¦\s]†_s`ba3Y2[n0Y$
z R']'kl_siiY ³ ´R $bdeY
ab]J_wn0`bmect³r$VS–$´k. M2g³r$VS–$´j. ³¦ –´ŠY2[Wcg0`UdŽ[Wm Z ³ ´+[W`ba 8 ( £ Yhc Y"p \|…dR… ­
³r$VS $´j.|³ ³ˆ ¶´J´q[rmegU_^gz
>…my[d‡Y!ab]J_wn0`bm‡c gg M2g¶³r$ S£$´j.|³ ³ˆ ¶´J´}Yf[rc
gw`bd[rm ? ³ ³¦ ¶´J´ 
( $
zÇȎc
[rm ³ ³ˆ ¶´J´/ $@…de_s]([Yhgµa3…]Jc†mykl`bdem‡Yh] ³ ´$ $-nw_^g3k cgŽ³ ^´/ $zRl}Y
abd‡`3[l? Ša3_^`b] c†_s`bc#V¢$ŠmedYh˜wmo[rcJY–n0Y2[#Yhp d"Yhp iYhgw? c([w!Yhc >c†Yhdy[®|`bY ³e\4 f´ Y"c
³ O4=!E´ ³¦ ¶´“zl}_sg3km‡dŽYh˜wmo[Wc†Yw!&; cJYhd@®|`bY$\:!=>G?c [W`ba |zl'_^g3k ,“[r`ba $
¸
Yhckl_^iÅiY'[r`ba &8 £
3deY[WYh`bd7c†Yh]JiY gg M2g4A N g M³W$VS>$´k. ³¦ –´Šgb_^gU€¦gw`bd7Y2[rc'klY"de`bm
kl_s]†]†Yf[ra3_^g3nU…gwc…­ $3n0_^gwc}d‡Ykl_|Y klm‡Yhg0cqY2[rc  zl}_^g‘k
gg M2B
g A N g gg M2gBA N g M2g gg
klY®|`bmZab]J_^`b›|Yd‡Yd‡YhiiY|z
8P: : 8
. $ǂ@ )ˆ4u‚ „.wƒ™ * g N
±dŽYhg>]UYfp [r`bd‡c†Y®|`bY[WmZžWYa3_|[rYř N g gg M2g/A N g Za3_^`b] $bžE‹‰_^jUc†meY"g3[}nwY2[mon0Yhiŀ
a3_^cJYhg0c“[_^cJ«b_^\s_^g3…2`b˜ªn0Y Ž³r$´“n0_^gwcdy…#[r_^iÅiY>Y2[rc gg z Žd‡`3[\ÇYhp gYhp ](…2deYhiY"g0c"-[Wm
 8§$|žWY
gb_sc†Y$u ¾ u>do…ab]†_^žWYfklc†m‡_^g>nwY$ [W`b]pF.2 £uY"c”žWYkl_sg3[rmynZYh­ ]JY
deY
€ˆYhg3[rY"i!jbd‡Yh€W$ Á * N#nqYlp x3gbma3…2]
Á * NW+³ S¶´k. Q|³ Z 4 Z ´bU Z #vX M * ™ N * M * ³
S>$´j. ³
¶´
—^Ygb_^cJYn0YiYhm iY  * N#deY$}€ˆYhg3[WYhijbdeYh€ nqYlp x‘gbma3…]
 * Nn+³r$zS¶´j. ³e¶´M * ™ N * M * ³
S ¶´j. Q|³ Z 4 Z ´bU Z #vX
}d‡_^]([Š[W_^m‡c' a a 8š$
z —^Y›s…mo[Šk"…dokE`bdeYh]qdeYaU]†_wn0`bm‡c Á * N M g  * N z3—^Ya3Yh`U˜#Y2p kl]Jme]†Y
Á * NvM2g  * N ³ S¶´j. Q|³ Z 4 Z ´bU Z #vXM * ™ N * M * ³e S>$´j. ³e¶´M2gw5(
((h³W$yS a ´k. ³
a ´M * ™ N * M * ³e a S a ´k. Q|³ Z a 4 Z a ´bU Z # a X
Œ@_^iÅiY
™ N * M * ³
S>$´j. ³e¶´M2g³W$yS a ´k. ³
a ´M2g™ N * 9(5
5( š™ N * M * ** M * ³e S>$´k. ³
¶´M2g³r$yS a ´j. ³
a ´M * ** M * ™ N * Yhc'kl_siiY
³
S$´j. ³e¶´M2g!³r$9S a ´k. ³
a ´"
³
Sw a ´j. Q|³ Z 4 Z { ´`U Z # { a X
{ * g * žWYn0_^my[Š…h›|_^me]
? ³jQ|³ Z 4 Z { ´`U Z # { a X^´R
[r_smecb 8 { ,ao0Yhc
³kQ|³ Z 4 Z { ´U Z # { a X^´R { a
n0_sg3kf { a z „m‡g3…d‡YhiYhgwchJdeY@ab]†_wnw`bmec Á * N M g  * N Y2[rcgw`bdw[Wm Yhc- a gUY-[W_^gwc„a‘…^[
kl_sg^žW`b\^`Y2p [}nU…g3[$z3È c[rm a ] ]sbau_s`b]*¶$3…d‡_^]“[RYhggb_sc(…2g0c ¼ d‹‰mo[W_^i_s]†ab«Umo[riÅY
kh…2gb_^gbmy®¯`UYY"g0cJ]†YoF.fGYhc ] B ]O.f ]vbYhcŠYhga3_|[†…2g0c
X
³ ] S ¶´k. Q|³ ¼ ³ Z ´4 Z U Z dvX
¸
+³ S ] ´k. Q|³ Z 4 ¼ ³ Z ´bU Z dvX
med Y2[rc'kEdo…m‡]'®¯`UY XM * X  * Yhc'®|`bYX M * Xf  * zl}Yabde`‘[
Á N Á * N XM
* *
Yhcqmed [r` ÅcŠnwYkl_sg3[rmyn-Yhp ]JYh]@deYkl_s`babd‡Y
³e,4(´R…­ kl_^gsžW`b\|…2mo[r_sgab]"Y2­ [@a3…]'$zb—^Ya3Yh`b˜Ån0_sg3k
[r`Uaba3_|[rY"] B,aobYhc'nb…2g3['klYkh…s[
Á * NvM2g  * N ³ S¶´j. Q|³ Z 4 Z ´bU Z #vX`M * ™ N * M * (5
{ !$# * *
(("³
S¶´j. Q|³ Z 4 Z { ´`U Z #vXM * ™ N * M * ³
S ¶´j. Q|³ Z 4 Z ´`U Z #vX
l}Yabd‡`3[l^žWYa3Yh`b˜ÆYfp kl]†m‡]†Y
™ N * M * ³e S¶´k. Q|³ Z 4 Z { ´bU Z #vXo+³
S¶´j. Q|³ Z 4 Z { ´bU Z dvX M * ™ N *
klY®|`bmZ]UY2p [r`bd‡c†Ya3…2]ŠYh˜0Y"iabdeYn0`deY"iiY'[r`Ume›s…gwchœ
,-.qqµ/
. tŽ4  57‚ ƒh4357ƒl&(13/243@5 @.675 13/24u!5 @.$ (Ç5. !57
‚ ˆƒh4qµ43'/ !!¦ƒ(qµ.
7.$~ƒh57/!57‚#13/43!
5 @.$ba :@/
2ˆ43/ƒ
gg M2g³r$VS>$ a ´k. Q|³ 4 w³Z´†´ U ¶$&Xi+³W$yS>$ a ´j. Q|³ 4 w³Z´†´U ,–$cXM2g gg g
A N M2g³W$yS>$ a ´j. Q|³ 4 w³Z´†´U ,–$cXo+³r$zS>$ a ´j. Q|
³ 4 0³Z´†´b
U ,¶$&XM2g A g N X
ȎgY=A YhcŠdy…
ab]†Y"imYh­ ]JY'…^[J[rYh]Jc†me_sg]UY2p [r`bd‡c†YnwYklY®|`bY
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(5 +³r$zS>$ a ´j. Q|³
u4 0³euZ´†´bUOu, X
}d‡_^]([Š®|`bY
³r$VS>$ a ´k. Q|³ 4 w³Z´†´ U *¶$&X`M2g ³W$yS>$´j. ³ 0³ˆ ¶´†´J5(
((+³r$VS $ a ´k.|³jQ|³ 4 0³´J´U F–$cX( ³ w³¦ ¶´J´†´R•³r$VS>$ a ´j. Q|³
u4 w³
uZ´†´U2u* vX
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³r$*Sq$´k. M g ³W$*S-$ a ´j. Q|³ 4 w³Z´†´bU ,–$cXo+³r$*Sq$ a ´j. Q|³ 4 0³´J´U ,–$cXYM g ³r$*Sq$´k. Ȏg`bcJmed‡mo[†…2g0cklYd‡YhiiYŠYhcŠd‹‰_s]†c†«U_^\^_sg3…d‡mecUY
p nwY2[Š™ N * bau_s`b] £^žWY›|_smo[Š®|`bY
Á * N`M2g  * N ³ PS ¶´j. Q|³ Z 4 Z ´U Z #vXYM * ³ePSR¶´j. Q|³ Z 4 Z { ´bU Z #vXYM * (5
{ ! # * *
b
( (J™ N * % M * ™ N * M * ³e S ¶´k. Q|³ Z 4 Z ´bU Z #vX
YhcklYhc†cJY{ [r_siiYŠY2[Wc-gw`bded‡Y[J…`0’ [rmm‡d3Yh˜0my[rcJY u, cgŽ³
¶´Çc†Yhd®¯`UY { šFayb…`‘®¯`bY"dkh…s[
³
4(´ +³e,4(*ae´(z
l}_^g‘k¶deY>ab]J_wn0`bm‡c Á * NJM2g  * N Y2[rcgw`bd[rm³e,4(´ÅYhc>³e,a 4(Fa‡´!gUY£[W_^gwca‘…^[
kl_sg^žW`b\^`Y2p [Ra‘…]$
z
—^YauY"`b˜n0_sg3k
[W`baba3_|[WYh] B,a3Yhc' šFay0Yhc'…2de_^]“[
Á * N Â * Nn
³ PS ¶´j. Q|³ Z 4 Z { ´bU Z vXYM * ³ PS ¶´j. Q|³ Z 4 Z ´ U Z #vXYM * ((
{ '!$# * N *
(((™ N * M * ³e S ¶´k. Q|³ Z 4 Z ´U Z #vXo9((
(( ³
³ S ¶´j. Q|³ Z 4 Z { ´`U Z #vX^´M * ™ ? *
{ '! # * N *
i_2¬|Yhgbg‘…gwc”d‡YdeYhiiY'[r`Ume›s…gwc
,-.qqµ9
. 1\t-14 Eŏ 57‚šƒh435„ƒl&“13/43!5 @.>‚„43/Oqr 2 75 13/243@5 @.#
(q5. IF.2 :
2ˆ43/ƒ
/
³ S¶´k. Q|³ Z 4 Z ´bU Z #vX`M * ™ N * š™ ? * M * ³ S ¶´j. Q|³ Z 4 Z ´bU Z #vX
.,
q .q;.
³ S ¶´j. Q|³ Z 4 Z ´bU Z #vX M * ³e S ¶´k. Q|³ Z 4 Z ´bU Z dvXi•³ S ¶´j. ³ ¶´
ȎgY=A Yhc"0a3…2]nqYlp x3gbm‡c†me_sg
™ N * ** M * A N * 9h
³jh  ´ ** M * ³
S*¶´j. %
% !
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gb_s]†i…2`b˜nwYo£z l
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do…’ˆ_^]JiY³
SK¶´k. ³¦ ¶´“„a3_^`U]
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³ S ¶´j. Q|³ Z 4 Z ´U Z #vX M ³
S ¶´k. ³¦ ¶´@Yf[rc-gw`bd[rm Z ³kQ|³ Z 4 Z ´bU Z dvX^´ 
g‹ Y2[rcŠa3…s[}kE_^gwc†Yhgw`nb…2g3[ ? ³ * ³ˆ ¶´J´"# £bY"c'[rmegU_^gbm‡dY2[rc Yhp \v…dq… ­
³ S ¶´j. Q|³ u46uZ´bUYu* X
l}Yabd‡`3[
³ S¶´j. Q|³ u46uZ´U2uF> X M * ³e S¶´k. +³ S¶´j.v³jQ|³ u46uZ´U2u* vX( ´
0
b
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Q|³ u46uZ´U2u* vX 9Q|³ u4VSb´U2u, 4fuZS >? ¶[W`ba X ³ –´+ Q|³ Z 4 Z ´`U Z #vX
klY®|`bmZab]J_^`b›|Y®|`bY
³ S¶´k. Q|³ u 4u´ UYu vXM * ³
S¶´j. 95(
(( +³ S ¶´k. ³ –´M * ³ S ¶´j. M * ³ S*¶´k. Q|³ Z 4 Z ´U Z #vX
YhcŠdy…
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 U U e 46 ³kh  ´  U nU v4 ³jh  ´
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\^]J_^`ba3Y2[Š Gn0Y&£z
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Q|³ Z 4 Z ´ U Z #vX Q|³ Z 4 Z ´ U Z #vXo ³ ´
„_s`b]k"…dokE`bdeYh]  * NvM * Á * N}^žE‹ _^j3[WYh]†›|Y·®¯`bY
³
S ¶´j. Q|³ Z 4 Z ´U Z #vX M * ³ S¶´k. Q|³ Z 4 Z ´bU Z dvX ³
S¶´j. * N
®|`bmg‹‰Yf[rcŠ…2`bc†]JY®|`bYd±‹JYhp d"Yhp iYhgwc X * N#n0Yd‹²…gUgbY2…` ³e¶´(z3—b‹²…2m7n0_^g‘k
 * NM * Á * N]•³r$VS¶´k. ³
¶´M * ™ N * M * X * NKM * ™ N * M * ³e S>$´j. ³e¶´
R
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’ˆ_^]JiY³
S$¶´j. M
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'
X !
M * X * N MX ! + * C N MX ! ' '
'
med Yhg]UY2p [W`bdecJY®¯`bY
 * NKM * Á * NW•³r$VS¶´k. ³
¶´M * ™ N * M * ³e S>$´k. ³
¶´
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™ * g N ,U cgŽ³e,4(>´j.OnU ³W$yS¶´k. ³
¶´M * ™ N * M * ³e S>$´k. ³
¶´
—b‹²…2m7n0_^g‘k
 * NM * Á * NWV,U &g„³
4(´k.OnU ™ * g N
^¸
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™ N * ** M * A N * [r_^`‘[Rdy…’ˆ_^]†iÅY
™ N * ** M *
³kh  ´ ³
S¶´j.
%
!
XŽ‹JY"p dhYhp iY"g0c ** Y2[Wc'®¯`‘…gwc…
­ d‡`bm7kl_^ijbm‡g3…my[r_^gÅdemegY2p …m‡]†Yn0Yc†Yh]JiY2[-nwYdy…’ˆ_^]JiY
³
PS¶´k. ¦³ –´(ha3_^`b]7 a3…]“kl_^`b]“…gwc„d‡Y2[Z[W_^`3[W€¦\s]†_s`ba3Y2[@n0Y£zOl}Y@abde`3[E2[rm * ³^´R£
…d‡_^]“[
³
S*¶´k. ³¦ –´M * ³e S¶´k. +³
S¶´j.v³ ³¦ ¶´ ´
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p ]UY2p n0`bm‡c†Ytn0Y
d±‹ Yhg3[WYhi!jUdeY o*4 *•n0Y2['[r_^`‘[r€ˆ\^]†_s`ba3Y2[gb_^]Ji…`U˜n0Y kl_^gwc†Yhg‘…gwc'[rc†]JmoklcJYhiYhgwcq Y"c
kl_sg0cJYhg0`‘[Š[rc†]JmoklcJYhiYhgwc@nb…2g3[ ¶zZŒ@_^iÅiYnwYaUde`3[
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±dZYhg]UY2p [r`bd‡c†Y®|`bY
N
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N
** N N ³ Z´
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iKYhm iYŠi…2gbmYh­ ]JY|wkl_^iÅiY
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žE‹²…2m
N
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* N
* N
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* N M ™ ? M ³ ”´M * N M ™ ?
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** N N ³ ‘´" ¼ * NvM * * N M iM * N M * Á * N }d‡_^]([E^žWYa3Yh`b˜šY2p kl]Jme]JY
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5( * N M * N M ™ ? M H * N M
l}_sg3k
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l}YiKYhm iÅY
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iM * N M * Á * N M g ¼ * N M * N M * * N M * ™ ? * I((
(( M * N M ™ ? M * N iM * N n0_sg3k
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—b‹²…2mq…meg‘g [rmŠnqYlp x3gbm@`bgbY …ababd‡mog kh…2c†m‡_^g ** N N n0Y * N M ™ ? M Ž³ @´RM * N M ™ ?
nb…2g3[}™ * N M2g Ž³r$´M2g N ™ * N z N
XZY2[…2ababdemykh…cJme_sg3[ ** N Y"c ** N [W_^gwcn0Y2[}jUm‰žWY2klcJme_sg3[Rm‡g0›|Y"]([rYf[-d±‹‰`UgbYn0Yd±‹ …`bcJ]†Y|œ
YhgY=A Yhc"
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*
b
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¼ * NvM *
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l}Ydo…
iKYhm iYŠi…2gbmYh­ ]JY|
* N
* N
* N * N ³ b´" * N`M * Á * NM2g ¼ * NbM * * N M M * N M * Á * N M2g ¼ * N M * * N R']
* NvM * Á * NvM2g ¼ * NvM * * N M H * NKM * * NŠ™ ? * M * * N M /J(5
(( * N M * * N * N M ™ ? M H
Yhc
M * N M * Á * N M2g ¼ * N M * * N M * N M * * N M * ™ ? * M * * N 9((
(( M * N M ™ ? M * N M * * N YhcŽžE‹ …m7n0_sg3kjbmeYhg ** N N ** N N ³ b´" 3z
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,a 4(Fa ´'Yhc³
>»g 4(»|´n0Y [rm g >Y2[Wc
`bg•Yhp d"Yhp iÅYhg0c n0Y™ * g N M2*
g *M2g™ * N Yhc au`Ug Yhp d"Yhp iYhgwcqn0Y™ * N M2g*M2g!™ * N 
…d‡_^]“[
* N N ³ Z´M * N N ³ a ´R95(
* *
(( * N`M * Á * NbM2g M2g ¼ * N M * * N M * N M * Á * N M2g a M2g ¼ * N M * * N
>…my[
cM2g ¼ * N M * * N M * N M * Á * N M2g a cM2g ¼ * N M * Á * N M2g a 9((
5( M g ™ * g N M g a M g a
±dZ›wmeYhgwcŠnw_^g3k
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M * N
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Ȏg3n w³ ´
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N
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M g ³ ”´
™ * g N M g ³W$´M g ™ *g N * N * N YhcdeY2[kl_sg3[rmyn-Yhp ]“…c†m‡_^g3[
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0³´-
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