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!" !"#$%&#$' &(! )*+,+-#./01#$% &'( )*)%&'&+ &,- ./01023 4)+ &, 5673 89&: &8&,), ;9&< ∈ ! =+) ( ) = ∅3=+) 4)+ &, 5>73 89&: " ∈ ! &8&,), ;9&< (") = #$ ? @)+ &'( $<AB$&:9' 5>7$?C?3 -$,D,4BE,-F<G%&" ⇐⇒ % ∈ (") ⇐⇒ [% = # ∨ % = ]'5!"7(M) #HB:F+ I-(+F,9-(%D,- G !" #$%& ( ∈) ' ()$*+&! * ∈ !' ,-,.!./0,& 1!" #$%& + ∈) '+&* ⇐⇒ + ∈ ! & (∀% ∈))[%&+ J⇒ %&(]'K9&: ( ∈) ? #$% &'( )*)%&'&+ ./01023 -$LAC<) =L$,), * ∈ ! &8&,), ;9&< (*) = ,(- ) | - ⊆ (()$%$,- ,(- ) = ,M(- ) <BF+) , &<D<9&MN &,- ./0102 4)+ &,( =%9F,M $,- -$LAC<)+$% &, #$%&? O$,D,4BE,-F<G+&* ⇐⇒ + ∈ (*)'⇐⇒ (∃- )[- ⊆ (() & + = ,(- )]$⇐⇒ (∃- )[- ⊆ (() & [+ ∈ ! & (+) = - ]]$+$% &'( #− %$⇐⇒ + ∈ ! & (+) ⊆ (()

⇐⇒ + ∈ ! & (∀% ∈))[%&+ J⇒ %&(]'P) +$,*<BH<)N &:( -$%D,)$:( +H):FL&:( 9&,(M <B(+) $+A%F,)<N3 +=,D,-Q;(&+N&)N )*8<N &'N +$%*<)H'N &,- R<:AMF+&,N #$'? S)N +TM(,-F< 4)+ +9=M9<)N? a#$%($ )*+,*,$ 2&34,& 5,! -6"7 #508.7 0(659:6 M = )$!$& &36"! #508.7,.( ;<=>=? ,5,& #"! 856.6 "6 @AB .M !#"6.).!&3 ,. C43:8" D#,"0E7' #"! @FB1!" #$%& - ⊆) ' ()$*+&! + ∈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`abcd/3 4)+ &,( ,$,B, =LQ< =%9F,N $,- )=+(,$,)<B &+ 5!7X 5"!7 <B(+)=D<)9&%N3 +DDL B9:N =L$,)+ +]<]+)%&'&+ (+ 8C<) $+A+F<B(<) 4)+ &, $;N 5=+) +(7F$,A,ZF< (+ +(&)F<&:$B9,-F< +-&% &, $A%]D'F+ 9< %D' &,- &' 4<()=%&'&+? @)+ !"#$% &'()*"(+ ,-./0&1 23% 4(03.5) 67819% 50' -:;4 <!*$/% =>?

!"#"$%&! '( )*+,&!$! -!+ ."/$0.! !"#$%& '()*$%& #$% +,-.-/0 %123456 *67 7189 8:);0 1$% )'67<37=$:*5 )#7 >3$?@89*7#7 !"#$0 !"#%A B#7C 7C7=53(*7)#5 )5 '()*$%& #$% +,-.-/0 D7 %1$?DE#$%*5 12C#70 4F3G& 6H67G#53; *C5G70 (#6 E4$%*5 5168EI56 '21$6$ )%<'5'36*EC$#585)#9 J5:<$%&0 EC7 ):)#;*7 =%)6'KC 736D*KC0 'A81A0 '76 (#6 %1$8$<GJ$%*5)45#6'$1$69)56& 13$#2)5FC 1$% 7C7=E3$C#76 )L 7%#E& #6& 7156'$C6)*EC5& EC?C$65& *5 7C7=$32 )L 7%#2 #7 '7D$36)*EC7 7C#6'5G*5C7A M7 >3$@89*7#7 !"#$0 !"#% I5'7D73GJ$%C (#6 $6 )%<'5'36*EC5& 5168$<E& 1$% E4$%*5 '2C56 H5C 51;?352J$%C #7 71$#58E)*7#7A!"#&" '()*+,-" !"#$%& '()*+, -+. /01213 M )-+4 +!+5+ '"6% )74+8+%549& &)+!8:6&'( *% '"!+&+ )74+8+ 9-(*;4< %&=&'(-%#9> =%4 *!+#+7*% 49 9!+?=%5@+.*% )-:4 ./01 (-& '"6% )74+8+ %549& 9A4(B)./12+*%( A N 6HE7 5GC76 C7 7156'$CG)$%*5 '2D5 ):C$8$ *5 #$ J5:<$&(0! ) )#$C '()*$M '76 C7 H;8K)$%*5 (#6 '2D5 7C#6'5G*5C$ 1$% H5C 7156'$?CGJ56 ):C$8$ *L 7%#( #$C #3(1$ 5GC76 2#$*$A OE#$%*5P" ∈# ⇐⇒ ! " = "! E#)6 1$% # =W !" ∈ $ ⇐⇒ ! (∃ )[C1D( ) & " = (0! )]!"% (0! ) ⇐⇒ ! " ∈ !'76 71$*EC56 C7 H5GI$%*5 (#6 ; #362H7M = # ! $ ! % 5GC76 '()*$& )%C(8FC0(#6 E456 #;C 6H6(#;#7 #$% +,-.-/ '76 (#6 6'7C$1$65G #;C 13(#7); E'"6% )74+8+%549& &)+!8:6&'( *% '"!+&+ )74+8+ 9-(*;4FA>73785G1$C#7& #$C H5G'#; M 0 13$=7CK&2(") =

∅! 7C <67 '2D5 ):C$8$ ! " 6= (0! )! ! 7C <67 '21$6$ Q'76 51$*ECF& *(C$ EC7R ):C$8$ ! " = (0! )!237 #$ 2(") 5GC76 ):C$8$A S67 #;C 6H6(#;#7 #$% +,-.-/0 1737#;3$:*5 13K#7(#6 <67 '2D5 ):C$8$ ⊆ # 0 (0! ) ∈ $ '76 2((0! )) = T 7C " ∈ $ '762(") = 0 #(#5 " = (0! &) '76 2(") = & <67 '21$6$ ):C$8$ &0 5I $36)*$:0E#)6 K)#5 = & '76 " = (0! )0 (1F& #$ 7176#5G $ $36)*(&A U #585)#9& #$%+,-.-/ 5GC76 1$8: 718(& )L 7%#9 #;C 153G1#F);0'(( ) = (0! ( ))N )45#6'$1$G;); )#$C M #;& 13(#7);& V'2D5 ):C$8$ 5GC76 6)$18;D6'( *5'21$6$ ):C$8$ 7#(*FCW 5GC76 ; 717G#;); (#6 A&9 '"6%M ?)74+8+ (0! )> .!"#?$%& 94-&)-+&$59 )-+4 M 94"*%)9 )-+ (0! ) '9& '"!+&+ M ?)74+8+ 9-(*;4(0! &)A OE#$%*5 & = ! (1! *) | * ∈ +13$=7CK& (0! &) ∈ $ '76 5I $36)*$:0"% (0! &)X⇒" ∈ &X⇒ (∀,)[" 6= (0! ,)]X⇒" -∈ $ !H;87H9 '2D5 M ?*E8$& #$% (0! &) 5GC76 2#$*$ )#$C M A YC ("! ,) 5GC76 $#585)#9& H67#5#7<*EC$% J5:<$%& )#$C M '76. = ! '(*! (1! *)) | * ∈ ! !"#$% &'()*"(+ ,-./0&1 23% 4(03.5) 67819% 50' -:;4 <!*$/% =>?

!" !"#$%&#$' &(! )*+,+-#./01#$#% & '$()&*M +,+-,./01%2 #& .* +,#2(#&230+ #., 45&M 6(7,$8., (0! ")'+2 (0! #)9 a:% #;, +,</.=&)&/>2'? @'>/+(; A& M ! "!#$%&'( )* $ $ $ B %,,&&5)%$#2 +, "′ = (0! ") '+2 #′ = (0! #) %0,+2 #+ +,#2'%0)%,+ =&7 %'=/&(.=&5, (#&% #+ (5,&8+ "C #C '2 +, <@(&7)%& ⇐⇒ ! ⊆ "×# & (∀' ∈ "′)(∃( ∈ #′)('! () ∈ & (∀( ∈ #′)(∃!' ∈ "′)[('! () ∈ ]!#$#% ; (3%#2'&=&0;(; &(M ) #;* & (#&, M +8;<%5%29 D=&8&-2()&0 (3%#2'&6=&2?(%., (+, +7#? %0,+2 =%/0=8&'&2C '+2 %0,+2 3/?(2)& ,+ 4;)2&7/-?(&7)% #;,'+#E88;8; A#%3,&8&-0+B =&7 <+ )+* %=2#/@F%2 ,+ +=&4%03,&7)% 242$#;#@* #&7* )%#&7* %8E32(#&7* /;#&5* 7=&8&-2()&5*9 G2H +7#$ #& ('&=$C #& >7(2'$ %/-+8%0&%0,+2 & %I?* =+/+4&(2+'$* (7)J&82()$* +=$ #; -%,2'? <%./0+ =/&#5=.,K +, &

M %0,+2 '$()&* (7,$8., '+2 ; & =/$#+(;C <@#&7)%M |= & ⇐⇒ ! ; & +8;<%5%2 (#&, M ⇐⇒ &(M)$L @'>/+(; M |= & 42+JE1%#+2 +* M '%! ( ,$-).,* )/0 &1C +88E %=0(;* +* M ! "!#$%&'( -)( &1C +* M ,(2)'3'( -)( &1C '98=9C $=.* +'&5-%#+2 '+85#%/+9M939 -2+ 'E<% 1%5-&* '8E(%., % C ) '+2 'E<% 42)%8? (7,<?'; *C @(#.NOPQRST(%!)!*) ⇐⇒ ! (∀+! ,)[+*,U⇒+! , ∈% ]

& (∀-)[- ∈ )U⇒ - ∈% ]

& (∀')[' ∈% U⇒(∃.)[456(.) & (∀-)[- ∈ . ⇐⇒ -*']]; 'E=.* =%/0=8&'; =/$#+(; =&7 %'>/E1%2 $#2 &2 % C )C * +=&#%8&5, '$()&(7,$8.,9 V(#. %=0(;*WSOXOY(%!)!*) ⇐⇒ ! (∀/ )[(456(/ ) & / ⊆%)U⇒ (∃!. ∈ ))[/ = M(.)]]

⇐⇒ (∀/ )[(456(/ ) & / ⊆%)U⇒ (∃!. ∈ ))(∀- ∈%)[- ∈ / ⇐⇒ -*.]]!; =/$#+(; =&7 %'>/E1%2 $#2 &M %0,+2 '$()&* #&7 WSOXOY9 Z0,+2 =/&#E(%2* -2+#2* % C ) '+2 *C &2 &=&0%* )=&/%0 ,+ +8;<%5&7, ? ,+ );, +8;<%5&7,C %I+/#E#+2[%0#% +8;<%5&7, %0#% $32C &2 %/);,%0%* \(3%#2'&=&2?(%2*] #&7* (H @,+, '$()&M′@3&7, ,$;)+C '+2 )+10 %'>/E1&7, $#2 * M′ ,(2)'3'( -)( &2 % C )C * +=&#%8&5,'$()& #&7 WSOXOY^ M/&'5=#%2 +=H +7#E )2+ =/&>+,?* %/_#;(;C =&7 @3%2 +=8?'+2 3/?(2); +=E,#;(;C .* %I?*9!"#$" %&'()*+" 72)# 8-29*0 )*. :;5<5= M1 = %1! )1! *> 8?@2'(0 )2 ⊆%2 ⊆%1> 8 ( 9( A(9'?B0 2.!CB8/ *2> )D)*( E2)' '*2 (U⇒ '! ( ∈%2F !M1 |= NOPQRST(%2! )2! *2) & WSOXOY(%2! )2! *2)!)-)' * M2 = %2! *2! *2 '%! ( ,$@"9 )( 8-29*0 )*. :;5<5=G

!"#$% &'()*"(+ ,-./0&1 23% 4(03.5) 67819% 50' -:;4 <!*$/% =>?

!"#"$%&! '( )*+,&!$! -!+ ."/$0.! !")./12+*%( # $%& '& ()*+,-.) /012& 32% ,M2 )*'&% 435.,6 5-'378'9 /0:/)%'& )/&7;<)=5,-.) 32% >%& 4?<) ∈!29(∃" ∈ !")(∀#)[# ∈ " ⇐⇒ # ∈!2 & #$2 ]%@528 ∈!2# A /032&5; !"# &7;<)=)% 52,' M19 )B35,'

M1 |= CDEFGHI(!2& '2& $2)&/,- 5;.&*')% 32% -/?0J)% 4?/,%, "1 ∈ '1 2:2,%, 152) >%& 37& 2& # ∈!19#$1"1 ⇐⇒ # ∈!2 & #$2 (4&% &B,= , M1 )*'&% 435.,6 5-'378'9 -/?0J)% 4?/,%, 5=',7, " 2:2,%, 152)>%& 37& 2& #9 # ∈ " ⇐⇒ # ∈!1 & #$1"1(2)7%4?9 ,% !"# 4&% !"# .&K* 5-')/?>,'2&% &-23 /,- &/&%2)* ; !"# >%& 2,(,5.:', ∈!2 4&% 4?/,%, " #$%& '& ()*+,-.) 32% , M2 )*'&% 435.,6 2,- LHDMDN9 /0:/)% /012& '& O)O&%1P5,-.) 32% &' " ⊆ !29 232) -/?0J)% 4?/,%, )2 ∈ '2 2:2,%, 152) >%& 37& 2&# ∈!29 # ∈ " ⇐⇒ #$2)2&4&% 4&23/%' '& )/&7;<)=5,-.) 32% .3',' :'& 2:2,%, )2 -/?0J)%# Q/)%(R " ⊆!2 ⊆ !1 4&% , M1 )*'&% 435.,6 2,- LHDMDN9 -/?0J)% 4?/,%, )1 ∈ '1 2:2,%,152) >%& 37& 2& # ∈!19 # ∈ " ⇐⇒ #$1)1()0>&K3.)',% 52,' M19 )B&0.3K,-.) 2;' %(%32;2& 2,- LHDMDN >%& 2,' M2 52,M1P5=',7, )19 4% &-23 .&6 (*')% 4?/,%, )2 ∈ '29 2:2,%, 152)

M1 |= (∀#)[# ∈!2 S⇒ [# ∈ )1 ⇐⇒ #$2)2]](&/3 2,' -/,7,>%5.3 &-2R6 2;6 5J)2%4,/,*;5;6 /0,4=/2)% 32%(∀# ∈!1)[# ∈!2 S⇒ [#$1)1 ⇐⇒ #$2)2]]%T/,7,>*K,-.) 210& >%& 2, 2-J3' # ∈!2 ⊆!1U# ∈ " ⇐⇒ #$1)1&&/3 2;' !"#&

⇐⇒ #$2)2 &/3 2;' !"#&/,- 2)7)%1')% 2;' &/3()%+; 2,- !"## V, >)>,'36 32% 2, /,7= :'& )2 ∈ '2%4&',/,%)* 2;' !"# >%& 4?<) # ∈!2 ()*J')2&% .) 2,' *(%, 203/,# a !"!$3"%4%5 W2; 5J)2%4,/,*;5; 2;6 !"# ()' &77?+&.) 4&<37,- 2;' /03P2&5; # ∈!2# $)'%4?9 ; 5J)2%4,/,*;5; .%&6 X/0823>,';6Y /032&5;6 2;6 .,0BR6* ( 1& % % % & ) Z/,- )4B0?K)% 32% ; ,0%52%4R 5-'<R4; * %4&',/,%)*2&% &/3 2&&'2%4)*.)'& 1& % % % & [ )*'&% ; *(%& ; * ( 1& % % % & )#\] &-23 2, &/73 <)10;.& ./,0,=.) '& 4&2&54)-?5,-.) 435.,-6 .:5& 5)435.,-69 .:5& 5) 435.,-69 J0;5%.,/,%1'2&6 5) 4?<) 52?(%, 2;6 4&2&54)-R6 !"#$% &'()*"(+ ,-./0&1 23% 4(03.5) 67819% 50' -:;4 <!*$/% =>?

!" !"#$%&#$' &(! )*+,+-#./01#$#%&'&() &*+ ,-%&+,*+ ,*+ .%/#) 01&120(+321.(4 5#1 ,1-3$(#6.17 8(9-*:.(&* (;<) =%-#2.1 &*+ !"#4 !"$! %&'()*+! !"#$%& '() *+,-./ M !.0 1$%& 2*#&34/ 152 "6.-.72345#$6!7 4 >2&9 3&*.*7 01# ?2&9! = ! " | (∀# ∈ @A("))[¬89:(#)B⇒ # = ]' 0/32' 1C&#0(#.?C9C ,*+ ,6;#<=.562& 2!+ 6. 7 %,9) *-D21.( 1+&< &'C ?CEC*#1 2&* =-%F/'.1 ""!,$4 GDC1# (:0*/* C1 $(D;*+.( %&# ' 0/32' ! (DC1#.(&1F1&#0<7 01# %&# 6#1 038( 2:C*/* $7$ ⊆! B⇒$ ∈! %$'/1$< %&# * H+2#0%) 0%2.*) ! ?I(# &'C #$#%&'&1 &*+ JKLMLN 01# 1,% &* !""(DC1# ,-%&+,* &') -./0O 01# ,-*H1CP)7 * ! ?I(# 10-#FP) ?C1 3&*.*7 &* 4G:0*/' 1,%$(#;'7 (H%2*C +,3-I(# &*+/3I#2&*C ?C1 3&*.*7 ,*+ D29) C1 +,3-EI(# 1//3 01# D29) C1 .'C +,3-I(# 2&* 8(.(/#10% .1) ,($D*W 4 =3C&9) +,3-I*+C,*//3 3&*.1 2&*C 0%2.* &*+ JKLMLN M! &*+ !"#7 0# 1+&% ,*+ ,-?,(# C1 03EC*+.( (DC1# 52 %#-;5%>,.0-% 6;5 2!+?%&@; 6;/ !#.;A.>-%5;/ !2#2A#"B.0 ,6.5*+,-. !!4 Q ,/<-') 1,%$(#;' (DC1# 9) (;<)4>2&9 &(!%'%(% ) ⇐⇒ ! RLSTUKV(!%'%() & JKLMLN(!%'%() WXEYZ&¬89:( ) & (∀# ∈!)[# )∈ 'B⇒ # = ]' ,-%&12' ,*+ 1,1#&(D 1,% &1 !%'%(% C1 ?I*+C &#) #$#%&'&() ,*+ .1) (C$#1EH?-*+C7 01# ?2&9 * ⇐⇒ ! (∃!)(∃')(∃()(∃ )&(!%'%(% )' ,-%&12' ,*+ (0H-3[(# %&# +,3-I*+C 03,*#1 !%'%(% .\ 1+&?) &#) #$#%&'&()4],*$(D;1.( &'C * 1,% &'C +,%8(2' %&# +,3-I(# &*+/3I#2&*C ?C1 3&*.* 01# WF?EF1#1Z &1 1;#P.1&1 &') -./04 ^-1 1+&< ' * 1/'8(:(# 2( 038( 0%2.* &') -./0,*+ ?I(# &*+/3I#2&*C ?C1 3&*.*7 01# (#$#0%&(-1 2&*C M!7 $'/1$<

M! |= *+]+&% 2'.1DC(# %&# +,3-I*+CM!E0/32(#) !%'7 .#1 $#.(/<)M!E2+C8<0' ( 01#03,*#* M!E1C&#0(D.(C* 7 &?&*#1 P2&(M! |= &(!%'%(% )%01# (#$#0%&(-17

M! |= RLSTUKV(!%'%() & JKLMLN(!%'%(),3-17 1,% &* !"17 ' &-#3$1 M = !%'%( (DC1# 0%2.*) &*+ JKLMLN4 G,#,/?*C7M! |= ¬89:( ) & (∀# ∈!)[# )∈ 'B⇒ # = ]%,*+ 2'.1DC(# 10-#FP) %&#M |= ¬89:( ) & (∀#)[¬89:(#)B⇒ # = ]%

!"#$% &'()*"(+ ,-./0&1 23% 4(03.5) 67819% 50' -:;4 <!*$/% =>?

!"#"$%&! '( )*+,&!$! -!+ ."/$0.! !"Ω

Ω1

∅ Ω2

6

?

K

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!! !"#$%&#$' &(! )*+,+-#./01"#$#%&$'()* +$,%# -%. & 0 /01*%#. %&2 +$'/')'34/&5!1 0"1 & !1 = ! !2 & "1 = ! "2 6⇒!2 0"2#7.# %&2 #+-8*.9&: ;+'<0%';)* -%. '. →1: $1 *32#. & /=0/& #>),2 >#. ' %-2'?%'; "1: >#. +#$')'34? )* %# →2: $2 @.# %' "2A B ;+-<*/& %&? !"# )#? 832*.>C+'.' %1 1← $1 %0%'.' ,/%*!2 = ! !1 = ! ("1& %1)&>#. >C+'.# +$'/')'34/& ' )*%#9( %42 ("1& $1) >#. ("2& $2)A D+- %& 1#/.>E.8.-%&%# +$'/')'.,/*42: ;+C$=*. %2 2← "2 %0%'.' ,/%* %1'%2F C$# & ' *3G2#. +$'/')'34/& )*%#9( %42 ("1& %1) >#. ("2& %2): >#. #+- %&2 !"# >#. %&)*%#1#%.>-%&%# %&? /;2<E>&? = !: !2 = ! ("2& %2): C$# !2 0"2AHC<* ! ∈ A0 *32#. #@2-: *8$#.4)02' !"#$# I).# %$.C8# /%' VJ: #>-)& >.#2 *32#. )& *8$#.4)02' %& '()*+,-: >#. & /;2<E>& +$'/')'34/&? = ! *32#./;2<E>& ./'8;2#)3#? /%&2 >KC/& A0 #+- %' !$%A L$#: #+- %' "$-1K&)#&#$!'( +$'>(+%*. -%. ;+C$=*. '$./%.>-? %*K*/%E? ( +'; *32#. +$'/8.'$./%.>-?@.# %&2 = !: 8&K#8E @.# -K# %# !&" ∈ A0:! = ! " ⇐⇒ ((!) = ((")#M' +*83' %'; #2%.<*)*K.4)02'; >-/)'; *32#. & >KC/&G+&K3>' %&? A0 #+- %&2= !:

A ="# ((!) | ! ∈ A0#N0%';)* ) * ⇐⇒"# (∃!&")[) = ((!) & * = ((") & ! 0"]&)* *+3>K&/& %'; !"#: >#. %*K.>C '$3O';)* %'2 .'"/ ."0123,3$1%,4"# 5/ ,#4? %&2 %$.C8# A&A& A N# #2#P*$-)#/%* /Q #;%-2 )* %' -2')# %'; +*83'; %';A: +'; *32#. *+3/&? >#. & >KC/& /;2-K42 %'; R 8*2 ;+C$=';2 C%')# /%'2 AA !"$! )*+,-./! ISTUVWJ I01J 6 A 37"-1 8/ ,#& 0#9 :;<=<>? @#9 18-"#@#1373@1@$4#" 0+" .(AB ."0123,3$7% +& 020? 0# .C7%,- D@1$#'B& 01 8-1 0+".(AB .'"/0+0-&E2345#$6!7 A B .8.-%&%#G>K*.83 %'; A *32#. -%. @.# >C<* @$CP&)# " ∈ V )*/=0/& #>),2 → >#. >C<* >-)1' $ ∈ ":3A((("& $)) = (("& %) | % ← $&+'; 8*3=2*%#. )Q 02#2 #+K- ;+'K'@./)-5) ∈ 3A((("& $)) ⇐⇒ (∃! ∈ A0)(∃% ← $)[) = ((!) & ! = ! ("& %)]

⇐⇒ (∃% ← $)[) = (("& %)]#D+Q #;%- +$'>(+%*. #)0/4? -%. >C<* 3A()) *32#. /(2'K': >#. *+')024? & %$.C8#A *32#. >-/)'? /;2-K42A7.# 2# 8*39';)* %&2 .8.-%&%# %'; XYVZV[: ;+'<0%';)* -%. + ⊆ A >#. I#+- %'01J: *+.K0@';)* @.# >C<* * ∈ + >C+'.' %'2./)02' @$CP&)# !! ∈ A0 %0%'.',/%* I\J ((!!) = *A D2%.>#<./%,2%#? >C<*!! )* >C+'.' ./')'$P.>- #2%3@$#P' !"#$% &'()*"(+ ,-./0&1 23% 4(03.5) 67819% 50' -:;4 <!*$/% =>>

!"#"$%&! '( )*+,&!$! -!+ ."/$0.! !"#$% &'()*+(,$)-. /01'12/( %$ 3(3$)4516/( (0)0781% 9,) # - 6= !:⇒" ∩"! =∅. ;$) <)$ ;*=( ∈ # . #>- ∅ $∈ " ? @5,A% = ! ⋃

" | ∈ # ∪ ∅&'→" ( ⇐⇒ ! (∃ ∈ # )['→ ( ∨ (' = ∅ & ( = ) )]&9016 ,$ → ;$) ) (B%$) C 5&85C $;/DE ;$) 1 ,9%1E ,16 " ?F1 ,1%)5/8%1<'*GC/$ % /( 5&85C $;/DE →" ;$) ,9%1 ∅ $%D;() 0'1G$%4E 5,C% ;7*5C A0.;$) <)$ ;*=( ∈ # .(%& ) ) ="# " $09 ,C% #,(,')//8%C. ,$6,1,);D- 0'151/1BA5C

('& () ∈ % ×" | ' = (**'$. $09 ,C% !"# ;$) ,1% 1')5/9.$A(+(%& ∅)) = +(%& ,) | , "← ∅= +(%& ) ) | ∈ # = +(" ) | ∈ # HI %&&= #-J)$ %$ K(BL16/( ,C /1%$K);9,C,$ ,16 +(%). 601=8,16/( 9,) ,1 % ′ (B%$) ,1%)M5/8%1 <'*GC/$ 5,C% A0 /( 5&85C $;/4% →′ ;$) ,9%1 ,′. ,8,1)1 45,(".0% ′ ⇐⇒ +(") ∈ #-N09 ,C% !"# 0*7).$A(+(% ′& ,′)) = +(% ′& ,) | , ′← ,′

= +(%& ) ) | ∈ # &$09 ,C% 609=(5C?O'$. <)$ ;*=( ∈ # . = +(" ) = +(%& ) ).+(% ′)&;$) #/( (0B;7C5C ,16 '(- /01'12/( %$ (0)78L16/( ;*01)1 , ′← ,′ ;$) ;*01)$0'151/1BA5C / /(,$L2 ,A% (%& ) ) ;$) (% ′& , )P ;$) $%,)5,'9GAE. /( ,1 BK)1(0)&(B'C/$. <)$ ;*=( , ′← ,′ /01'12/( %$ (0)78L16/( ;*01)1 # ∈ # ;$) ;*01)$0'151/1BA5C 0# /(,$L2 ,A% (%& , ) ;$) (% ′& ,)? QR $6,8E ,)E (0)71<8E (B%$)(2;171 %$ (0$7C=(2516/( 9,) C 8%A5C1 =⋃

/ | ) "← ∅ ∪ 0# | , ′← ,′ ∪ (∅& ,′)(B%$) 0'151/1BA5C /(,$L2 ,A% % ;$) % ′. KC7$KD +(%) = +(% ′)?F(7);*. <)$ %$ K(BL16/( 9,) 1 ;95/1E A );$%101)(B ,C% $'&D ')'. 85,A "<'*GC/$ 5,1%A /( 5&85C $;/4%→$ ∈ A? J)$ %$ ;$,$5;(6*516/( K)$;95/C5C,16 " 5,1% A. $';(B %$ 1'B516/( 8%$% AM,(7(5,D Æ. ,8,1)1% 45,(A |= (∀) ∈ ")[Æ()) = Æ(,) | , $← )]&

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( →# .) = ! /' ∈- | (∀10 ')[10 ×# .]

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