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wGl3734>2/.6LHD?MJkP43,DC4 OY23Bk*ME9J,3T? K Yw BXDCFX@tM2343M=56 B L>MDN.?,3F2>MIKCOYMEM;,rh (TG.OY23Bk*ME9J,3T6. 3t U/4CDC4kPM s U/MX9DC4BG234B6,3T?M=432/.:0CB?TGB3h ]9DC4 yl3734323.6LHD?MJkP43 , |Uv;,/BBNfNB:V;2/.9o` |U 4CDC4 U ME9oDC4[23M l3734323.6LHD?MJkP43,rU£;,/B}B>fCB:V23.EfC∈B:V2/.TGBL>7>KC5G4CRBNfNB:V;2/.43u%LHDG`DGB?5G434NRBGl3M=73.?,/BG7dBGT3 ∃ U ∩ U ∪ 4,/.?0 L3.ED?MEMg-l37>4/LNKCO].ED qkPK6V:M=l>l3M M;.:23BT0HBG23n3M 3 J3TGM=03.>h3,>4NRBNfNB:V;2/.4NW `6RSU/7/.?TGMJ239=,>TGBIOY2>BEkPME9=,>TIOYBkP23BVE.?l>439;.?,>F,/.?0v
a ⇔ ∈ og ⇔ ∀ | ∈ w j 23BkPME9=,3TG. A4 P7/.?TG23_ ,/BG5XL3. 4,/BXDCFG0HB ,/BG5XL3.>U 0HBG5XL3. LHDG` DCAfCBG56B U ∈ 7/.?TG2>BN9J4CDCFG23B ∈ w3hCv2/.,323BC9=,3F8KC,3TGMJ7>k*L>M=234>u%azw?,/BG5XL3.4#,/BXDCFG0HB8,/BG5XL3.?w>UGwG7/.?T?23BN9J4CDCFG2>BGwHgih X^ · ³ 2 OY23BkPMX9J,3T6. 2/.XV;_qT6.?M;,>9o`[OY23BkPME9=,3T?B oU n>MoDC430HBGO 9;B6L>MJ7>k*.X*MEMX9`T |hv+-,/BIB>fCB:V23.EfCB:V;2/.MtTGTGMoLNM=23B+-732NW 9=,/BGO ^7SMoL>MJ7SB?OT é% gih/mD?BG5G43MX9J034NR BNfNB:V;2/.4C`6RSUsOYBEkP23BV.?l3439=.?,3F
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∪ { | ;,/BGu%ABG7>OK6D?M { | Ia Sg } BNfNBXV2/.B?TXD?M;,3TGB:W 7C`C*4NR a Sg w}a ,/BBNfNBXV2/.432/.6LHD?MJkP43,1.?2>,/BG7>K>gih -43OYTGBXDH_ ∨ 4 BXV2/.g ◦ ) BNfNB:V;2/.B:V43n>43 B6,/B>fG73.:kPMiW ^ )} §t $(23BG5XL3.eV;2/.7343OYM=2>M=232>KC TL>7NKN56B?Ol3BG7H`NL>0HM?s23*M va a Sg goU3. (a v a Sg gih ]9DC4 a Sg d4 2/.XV;_qT6.?M;,39`%OY2>BEkPME9=,>TGBTG9;M R ∈ US0HB6,/BG73_qMt`CTXDG`HA,39`B>fG7/.EVE.?OY4%0/.:034NRHW ,/B ∈ ih c 7 l3B6L>OY2>BEkPME9=,>T6,. + ⊂ WdOY23BkPMX9J,3TGB(TG9;M R8iDGMJOYM=23,/B?T ∈ iU l3M=7dM R6BGLH`6*4NR TJD?M=OYMJ23,3_ + p ; X^ · ³ 2 +" ⊂ #W#9;BGT?B:W 0NKCl323BC9=,>FTG9=M R%JD?M=OYMJ23,/BGT3U l3BXD6KNM=2323_-R0/.:0[BNfG7/.XV;_ iDGMJOYM=23,/B?T +"oh fG73.XV l3B6L>OY2>BEkPME9=,>T6.BNfNBXV2/.fG7/.EV
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q
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,/Bk*L>ME9=,>TGM=232>BNMPl37dMXBNfG7/.XV:BGTG.?2343M ;h l37SMoLNMoD?M=234>M|KC2303n343uyOY23BkPME9=,3TG.l37dBG4NV;TGMoL>M=2>4C`[l3734>2C`C,/BT TGMJ7S9J434#BN9J23BGT6.?2>43uO].?,/M=O].:,343034rU60HB6,/BG73.=`f6.XV;437>KCM;,39`23.,/MXB?73434OY23Bk*ME9J,3T3h L>7>KC5G4CR TGM=7d9=4H`6R BC9=23BGTG.?2343u^O].?,/MJO].?,3430>4 BC9=23B?TG23_qOb`CTXDG`CM=,39o`l3BG2H`C,343M |KC2303n34>4rUvVE.6L3.:2323BGu[ABG7>OK6D?BGuDC4>fNB0/.:0343OW,/B.DC56BG7343,>OYBGOqU .OY23BkPME9=,3T6. BGl37dMoL>MDG`CA,>9o`\T^,/M=73OY4>2/.XR |KC2303n>43urh (.:kPM}T .?0H9=43BGO].:,34343TGMoL>MJ2323BCMPT?_-@tMABGl37dMoL>MoD?M=2>43M1|KN2>03n3434 439=l>BXDCF:V=KNM=,39o`2/.?7C`CLNK 9YAB?73O].EDCFG2>B:W D?BG5G4>n34C`dU=V.6L3.?232/.J`ABG7>OK6D?BGu12/.£`6V;_q0HM (M=73OYMD?B:W 7dM=2>0CMDG`CwazT?_q9=0/.XV;_qT6.?,/MDCFG2/.J`|KC2303n34C`Ngih BEDHKC43w|KC2303n34>mT(9=OY_q9DGM£TG_-@*M=l37>43TGMoL>MJ2323BG56BB?l37SMoLNMoD?M=234H`POYBk*2>B3U TGBC9=l3BEDCF:V:BGT6.?TX@t4>9=FB6L>2>BGu4NVt.?0H9=43BGOazwG9oR6M=OYB?ul3BGL>9=,/.:23BGTG034>wHgih )% !A)%$ 8 $ !))%&A$ c r [ KH9=,3Ft2/.tOY23BkPME9=,3TGM VE.6L3.?2>B9;BCB6,323B:@*M=2>43M ∼ UHK6L>B:W TXD?M;,3T?BG7C`CtMEM19oD?MoLNKCt4>OKH9oD?BGTG4H`COqh t 2 |D?`DHAfCBG56B ∈ |U ∼ h ^ G 2 |DG`#DCfG_-R ∈ qU a ∼ >g ⇔ a ∼ Sg ^¶}<r
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4 7/.?TG2>_ ,/BG5XL3.4,/BXDHFG0HB,/BG5XL3.>U 0HBG5XL3.BG2>443OYM=,B6L>2>44,/MIkPM iDGMJOYM=23,>_8 a ∀Q a ∈ ⇔ ∈ g g ⇒ T 4 9iKH*ME9=,>T?KCM;,OY23BkPME9=,3TGB ^ µ -DG`1DCfG_-ROY23BkPMX9J,3
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9=T?BGu39=,>TGB3Ud0CB6,/B?73_qO OYBEkPM;, BNfXDN.6L3.?,>FOY23BkPMX9J,3TGB h -DG`P0/.:k*L>B?56B8OY23Bk*ME9J,3T6. 9iKH*ME9=,>T?KCM;,Il>B6LCW OY23BkPME9=,3T?B ⊂ }UH9;BC9=,3.?TXD?M=232>BNM4NV8TG9=M R ∈ }U6K6L>B?TXD?M;,3TGBG7H`C(*4NR 9JTGBGu39J,3T?K h } BNf Mo^ L>43¶23MJ23 43 M T -DG`eDCAfG_-R^OY23Bk*ME9J,3T 4 9iKH*ME9=,>T?KCM;, ∪
∪ { | a ∈ 8g ∨ a ∈ g } G^ } [} ,3OYM;,343OqU{BEkPME9=,>T6. 9iK6tME9J,3T?KCM;,OY23BkPME9=,3TGB MJ56Bl>B6L>OY23BkPME9=,3T 0 h G^ } [ 8BG23M=23_qM#OY23Bk*ME9J,3T6.T.?0H9=4>BGO].?,343MX9J0HBGu,/MEBG7>434'OY23B:W kPME9=,3T19=,>7SBX`H,39o`9oD?MoLNKC(*43OyB>fG7/.EV?B?Oqh j 23BkPME9=,3TGBP4NV2>KGDG`PJD?MJOYM=23,/BGT8
∅ h j 23BkPME9=,3T?B*4NVB6LN23BG56BJD?M=OYMJ23,/.
,/B {∅} UGOY23BkPME9=,3T?B*4NVAL>T:KHRtJD?M=OYMJ23,/BGT
,/B {{∅} ∅} OY23BkPME9=,3TGB4NV*,37dM RJD?MJOYM=23,/BGT#W
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23MBNfX`6V.?,/MoDCFXW 23B[OY23Bk*ME9J,3TGB3h -0/.:kPMJOqU{BNfXDN.?9J,3F[BGl37dMoL>MoD?M=2>4C`e,/Bk*L>ME9=,3TGMJ2323BGuTG_q9=0/.EV_-W T6.?,/MDCFG23BGu |KC2303n34>4 a >g ⇔ a >g ,/BT?9;MtOY23BkPME9=,3TG.>US2>B9;BGT?BG0NKCl323BC9=,>FTG9;M ROY23BkPME9=,3T2>M*BNfG7/.XV=KNM=,%OY23Bk*MoW 9=,3TG.a r.?OYM=t4C `d¨UG. tWvOY 23BkP MX9JL,3 TGB3UXTG 9;M]J D?M= OYKHMJ9=2N,3W F ,3_ 0HB6,/BG7SB?56B}D?MJk*.?, T}BNfXDN.?9J,34 BGl37dMoL>MoD?M=2>4C` h-BG5XL3.9JK6*ME9=,3T?KCM;, OY23BkPME9=,3TGB U>9;BC9=,/.:TXD?M=2323BCM84NVPTG9=M R dU3LHDG`#0CB6,/B?73_-RTGMJ7323B a >giUN9 0/.?0>43OW,/B ∈ h ^ )} w {R6M=O].l>B6L>9=,3.?23BGTG0>43w'
23M.?0H9J43BGO].>U£.wGl>7/.?TG4CD?BTG_-W TGB6L3.:w3US0HB6,/BG7dBCMtl3B03.:k*L>BGuTG_q9J0/.XV;_qT6.?,/MoDHFG23BGu'|KC2303n34>4[9J,37dBG43,9JTGBG .?0CW 9=43B?OKSh $23.EM56BGTGB?7C`dU]w6.?0H9=4>BGO]. w',/B9=23_qu 2/.?fCBG7.:0C9J43BGOqU{0/.:k*L3.J` VE.6L3.:232/.J`9JTGBCM=u TG_q9=0/.EV_qT6.?,/MDCFG23B?¥u|KC2303n34>M=urh KH9=,/BCM[OY23Bk*ME9J,3TGB 9;B6LNM=7>kP43, JD?MJOYM=23, ∈ m,3.?0HBGurU/h tBG5XL3. K KH y9=,>ME9=F ,3 F −→ 9= 73FGMJ03,343}UTG2>vBN,/MBB6ME,/9=,>B>Ff?7/B6.:,/kPBNM=fG237/4>.XM W } kPM=2>43M −→ }U K6L>BGTEDGM=,3TGBG7C`H(*MEM ◦ −→
]h $2/.BGOK}JD?MJOYM=23,>K T9;BCB6,3TGM;,39J,3TG43M23M=034>umJD?M=OYMJ23,^4NV l37dBCB>fG7/.EVE. a >gih ¤A0H9=4>BGO].yTG_qfCBG7/.yTEDGMJBG0H9; .EDHFG23_-R^9D?MoL>9=,3T?43urh .?l3734>OYM=7rU 4NV23MEM9oD?MoLNKCM;,rU{43234>BG0H9 -.?2/.XR>.XW .?7d9=0HBG56BHgoh:¤0H9=43BGO].T?_qfNBG73.23MoVE.?TG4>9=43O].AB6, BC9=,/.EDHFG23_-R.?0H9=43BGO (M=73OYMDGBXW 7dM=2>0CMDG`dMEMI23MoDCFXVJ`234L>B?0/.XVE.:,3F3U2>4BGl37dB:W TGMJ735G2>KC,3FT,/BGu'9J439=,/MJOYM.?0H9J43BGO ac,/B L>B?0/.XVE.D BXD8B:2vUr439=l>BXDCF:V=K6` ^ BG2/.I 23 M=l3t7d B6 ,343 T?BG7SgiMJh#MJ23OY43Mo4rD?B:UvW T6Ud.>U30/,/.?0B ¨ 4[Tl37dMoL>l3BXD?BkPM=234>4rUQ,/B.?0H9=4>BGO].TG_qfCBG7/.23MJTGM=7323.ac;,/B%L>BG03.XVE.ED%xsMoL>MoDCF6gih BXDCF:@*43239J,3TGB.EDCFG,/MJ732/.?,>43TG23_-RtTGM=7d9=4>uBN9J23BGT6.:2343utO].:,/M=O].?,34>034az0HBG2>9=,37>KC0NW ,343T?4NVOqUG4>23,>KC43n343B?234NVOqU6KGDCFG,>7/.?{432343,34CVO{gY23Mql3734NV;2/.?M;,I.?0H9=4>BGOKTG_qfCBG7/.4 MEM*9D?MoL>9=,3T?43urhS fCBXDCF:@*43239J,3TGM17/.XVL>MoD?BGTIO].?,/M=O].:,343034fCMiV.?0H9=4>BGOY_ TG_qfCBG7/. OYBkP23B#B>fCBGu3,3439JF3U3439Jl3BXDCFXV;KG`T?OYMX9J,/B.?0H9=43BGOY_ TG_qfCBG7/.0/.?0CKNW,/B#BC9oDN.?fEDGMJ2NW 2>KCZTGMJ7S9J43a~9=Oqh h h goU:23M£TXD?M=0CKHPKCZl/.:7/.6L>BG0H9;BGT>h BX,/BGOK*OY23BG5G4>M-KCFA.?0H9J43BGOY_TG_qfCBG7/.4tMEM-9DGMoLN9=,3TG4>urU?4CDH4rUEl3BA0373.?u323MJuOYMiW 7dM?U/B6,3OYMJU " !1%2"3 465 %-,H2%+!:;1%=< @#%+ "&I+# b e2%M a9b * BI c "
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" A 3l " A. l " " A l V )S` ) < .f`9. A >D % A ' Q ' [- SO-SO ( l ( $85 -' ( @8 ' 3_ ' R '@i J ) ' '7R T >g " #O "0 ; F V O R '@i V ' SO )/ '.' SO = e#O T R R O"-[ T ' F SU A` ' $ ' .' (O]l \[ - 43R ' R ; R ; 5 V ' SO ) ' ' R, ('[ ' 0 V 1 -- ' 5 '- @' 3 ' O;B A10 − 3l ) 2 ∈ ⊂R & )G0B` . V ) = % A R ' =R $ '3' ' A l ≤ 6A.3l 6A l @' ( ' ?$W[ ; i & ($1 ,R T ;.: =i [O -O =i V O $ R
$ ;4T [ S2O,& ) ) = % A Q ' ( ' $W[ ; i $ /R ' √ " 0@V -[ S-O p AWA 6A l l Q ' $ 43 ) @' ( ' $W[ ' O (($ ' '@i SO Q ' 5SO $2&
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≤ 7 8" ; ;!( $ 1 O R 34 ' @' '@R 5 Rb' #O P Z V O7E ' & R i5 -' @' )3 ' .: \[ 3 T'3 M A l 07?V Q'T m⊂Z ' \[ O R 3 '@' ( ) ' -- ' AC O ' $ ( $P '. 5 ' & m Z ] ) I [ O # 5
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Gl V ' (O ) = % A Q ' ' $W[(; i ($hO- ]'@R ' ? O V = 8" ; ;!($ 1 O R 34 \'@' '@R 5 Rb' O Q ) O 5 m " { [
} 1 O =)' ' ( Rb' T i T V ' SO )' ' ;.: ( ; ^SO#V ' F : & ( 4' ( ' -&0)7 ' 6A l - P (O ' ]' ) V V " " 6
=! 8" ; ;!($ 1 S O ' '@R 5 R ' O Q )/ ' 6A l ≤ 07V ' & SO )7 ' 6A.3l A.3l 0 1?c Q ' ]' ' ]' ' V O 1@' '@i ( ( & > = % A Q ' ' $W([ ; i ($% /R SO f (3 ( ?bV =! 8" ; ;!($ 1 S O ' '@R 5 R ' O Q )/ ' 6A l ≤ 07V ' & SO )7 ' 6A.3l 0 c Q ' ]' ' ]' ' A` V V \ O R 34 ' @' l V ≤ > = % A QeT \[ √ " 0 '\' ' " ' 5V Q ' "& ' ?$W[ ' O ($] R, T ;.: = i [O -O =i )= ' ?; A. l ≤ 0B =! 8" ; ;!($ ' 1 ' ' #O $ ' $ ' R O ' SO $ T'3 ( O J A=g ' \' O $ ' ( $ ' [ 4 $0 ' PO ' Q T* ' (5T$ )7 1 '
$SO : ( T'38 (3O23(l V /R ' ' *)7 ' '7R 54& Rb ' ' O V % SO Q O R 3 ' @' V ' SO ) ' A.'l 1 ' \' O $ ' ( $ > = % A #O (EO %4@' '@i (( ( ,) ' SO )= ' 6A l ≤ 6A l J A l ≤ 6A l J oprqtstutu\u(vxwzyzw|{tuS}t~!~}t~x~utu
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% A ) ` ) = % = ` ) . [ -- ' ]' )e ' '@R 5 R ' O Rb'. ' O& 7!2 8" ; QeT \ ( ' R, =R T '@i ;4 SO ) ' '@R O ; \' @'7R 6A l 6A l 6A l V ' SO ) ' S ' ?;SO (RiS O 7→ | | SO '@i & ' ' 4" & (SO# T - ' V7Q'T R OI[- -9 =R, [ 6A l V 7! 8" ; '!S O 43 - ' ; 7→ | | [IO O '7R 5 Rb' & O (O Q TS ' .;4P ' 4;4c 3O7E( (O cR '@R 5 Rb' #O i SO $ ?c Q 4' '@R & 5 Rb' O ' ?;SO (SO# $ &O ( ']' '@R 5 R ' O Q Q ' ' & O \OWE( SO 8 SOI-[ T O ( A ` )7B2>=6Pp&fqE7ED3+(+&fPc5 ∈ C!AB& 0 Bo C 5+2e7 .10 5'7B? 0 4