:: HALLMAR1 semantic presentation begin theorem :: HALLMAR1:1 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "X")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Y")) ")" )))) ; scheme :: HALLMAR1:sch 1 Regr11{ F1() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set F1 "(" ")" ))) "holds" (Bool P1[(Set (Var "k"))])) provided (Bool "(" (Bool P1[(Set F1 "(" ")" )]) & (Bool (Set F1 "(" ")" ) ($#r1_xxreal_0 :::">="::: ) (Num 2)) ")" ) and (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set F1 "(" ")" )) & (Bool P1[(Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1))])) "holds" (Bool P1[(Set (Var "k"))])) proof end; scheme :: HALLMAR1:sch 2 Regr2{ P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool P1[(Num 1)]) provided (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool P1[(Set (Var "n"))]) ")" )) and (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool P1[(Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1))])) "holds" (Bool P1[(Set (Var "k"))])) proof end; registrationlet "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) ($#v2_relat_1 :::"non-empty"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) "F") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F"); end; theorem :: HALLMAR1:2 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; registrationlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set "A" ($#k1_funct_1 :::"."::: ) "i") -> ($#v1_finset_1 :::"finite"::: ) ; end; begin definitionlet "F" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "J" be ($#m1_hidden :::"set"::: ) ; func :::"union"::: "(" "A" "," "J" ")" -> ($#m1_hidden :::"set"::: ) means :: HALLMAR1:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "j")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) "J") & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A")) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "A" ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" )) ")" )); end; :: deftheorem defines :::"union"::: HALLMAR1:def 1 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "J")) "," (Set (Var "b4")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool "ex" (Set (Var "j")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" )) ")" )) ")" )))); theorem :: HALLMAR1:3 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Var "F")))))) ; theorem :: HALLMAR1:4 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "J")) "," (Set (Var "K")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "J")) ($#r1_tarski :::"c="::: ) (Set (Var "K")))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "K")) ")" ))))) ; registrationlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "J" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_hallmar1 :::"union"::: ) "(" "A" "," "J" ")" ) -> ($#v1_finset_1 :::"finite"::: ) ; end; theorem :: HALLMAR1:5 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; theorem :: HALLMAR1:6 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set ($#k2_tarski :::"{"::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k2_tarski :::"}"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" )))))) ; theorem :: HALLMAR1:7 (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" )))))) ; theorem :: HALLMAR1:8 (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set "(" (Set (Var "J")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ))))))) ; theorem :: HALLMAR1:9 (Bool "for" (Set (Var "J1")) "," (Set (Var "J2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set "(" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "J1")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "J2")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set "(" (Set (Var "J1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "J2")) ")" ) ")" ")" ))))))) ; theorem :: HALLMAR1:10 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))))) ; begin definitionlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"Cut"::: "(" "A" "," "i" "," "x" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F") means :: HALLMAR1:def 2 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A")) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool "(" "(" (Bool (Bool "i" ($#r1_hidden :::"="::: ) (Set (Var "k")))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "A" ($#k1_funct_1 :::"."::: ) (Set (Var "k")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) "x" ($#k1_tarski :::"}"::: ) ))) ")" & "(" (Bool (Bool "i" ($#r1_hidden :::"<>"::: ) (Set (Var "k")))) "implies" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "A" ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ")" ) ")" ) ")" ); end; :: deftheorem defines :::"Cut"::: HALLMAR1:def 2 : (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b5")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" )) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b5"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "k")))) "implies" (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ")" & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "k")))) "implies" (Bool (Set (Set (Var "b5")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ")" ) ")" ) ")" ) ")" )))))); theorem :: HALLMAR1:11 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set "(" ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))))))) ; theorem :: HALLMAR1:12 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set "(" ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) "," (Set "(" (Set (Var "J")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set "(" (Set (Var "J")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" )))))) ; theorem :: HALLMAR1:13 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set "(" ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) "," (Set (Var "J")) ")" )))))) ; theorem :: HALLMAR1:14 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "J")))) "holds" (Bool (Set ($#k1_hallmar1 :::"union"::: ) "(" (Set "(" ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) "," (Set (Var "J")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set "(" (Set (Var "J")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ))))))) ; begin definitionlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "X" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "A" be ($#m1_hidden :::"set"::: ) ; pred "A" :::"is_a_system_of_different_representatives_of"::: "X" means :: HALLMAR1:def 3 (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "F" "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) "A") & (Bool (Set ($#k4_finseq_1 :::"dom"::: ) "X") ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set "X" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )); end; :: deftheorem defines :::"is_a_system_of_different_representatives_of"::: HALLMAR1:def 3 : (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r1_hallmar1 :::"is_a_system_of_different_representatives_of"::: ) (Set (Var "X"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "F")) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "X")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ")" )))); definitionlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); attr "A" is :::"Hall"::: means :: HALLMAR1:def 4 (Bool "for" (Set (Var "J")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "J")) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A"))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "J"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k1_hallmar1 :::"union"::: ) "(" "A" "," (Set (Var "J")) ")" ")" )))); end; :: deftheorem defines :::"Hall"::: HALLMAR1:def 4 : (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) ) "iff" (Bool "for" (Set (Var "J")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "J")) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "J"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k1_hallmar1 :::"union"::: ) "(" (Set (Var "A")) "," (Set (Var "J")) ")" ")" )))) ")" ))); registrationlet "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) "F") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_hallmar1 :::"Hall"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F"); end; registrationlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) "F") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_hallmar1 :::"Hall"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F"); end; theorem :: HALLMAR1:15 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v2_relat_1 :::"non-empty"::: ) ))) ; registrationlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_hallmar1 :::"Hall"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_relat_1 :::"non-empty"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F"); end; theorem :: HALLMAR1:16 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) )) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 1))))) ; theorem :: HALLMAR1:17 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) )) "holds" (Bool "ex" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "X")) ($#r1_hallmar1 :::"is_a_system_of_different_representatives_of"::: ) (Set (Var "A")))))) ; theorem :: HALLMAR1:18 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool "ex" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "X")) ($#r1_hallmar1 :::"is_a_system_of_different_representatives_of"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) ))) ; begin definitionlet "F" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); mode :::"Reduction"::: "of" "A" "," "i" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F") means :: HALLMAR1:def 5 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A")) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A")) & (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) "i")) "holds" (Bool (Set "A" ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" ) & (Bool (Set it ($#k1_funct_1 :::"."::: ) "i") ($#r1_tarski :::"c="::: ) (Set "A" ($#k1_funct_1 :::"."::: ) "i")) ")" ); end; :: deftheorem defines :::"Reduction"::: HALLMAR1:def 5 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "," (Set (Var "i"))) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) ")" ) & (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) ")" ))))); definitionlet "F" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); mode :::"Reduction"::: "of" "A" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "F") means :: HALLMAR1:def 6 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set "A" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) ")" ); end; :: deftheorem defines :::"Reduction"::: HALLMAR1:def 6 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A"))) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) ")" ) ")" ))); definitionlet "F" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); assume that (Bool (Set (Const "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Const "A")))) and (Bool (Set (Set (Const "A")) ($#k1_funct_1 :::"."::: ) (Set (Const "i"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; mode :::"Singlification"::: "of" "A" "," "i" -> ($#m2_hallmar1 :::"Reduction"::: ) "of" "A" means :: HALLMAR1:def 7 (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" it ($#k1_funct_1 :::"."::: ) "i" ")" )) ($#r1_hidden :::"="::: ) (Num 1)); end; :: deftheorem defines :::"Singlification"::: HALLMAR1:def 7 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "i"))) "iff" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" ))))); theorem :: HALLMAR1:19 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "," (Set (Var "i")) "holds" (Bool (Set (Var "C")) "is" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A"))))))) ; theorem :: HALLMAR1:20 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ) "is" ($#m1_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "," (Set (Var "i"))))))) ; theorem :: HALLMAR1:21 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ) "is" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A"))))))) ; theorem :: HALLMAR1:22 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "B")) "being" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "C")) "being" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "B")) "holds" (Bool (Set (Var "C")) "is" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A"))))))) ; theorem :: HALLMAR1:23 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "B")) "being" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "i")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))))) ; theorem :: HALLMAR1:24 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v2_relat_1 :::"non-empty"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "B")) "being" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "i")) (Bool "for" (Set (Var "C")) "being" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "B")) "," (Set (Var "j")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Set (Var "C")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "j"))) & (Bool (Set (Var "C")) "is" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "i"))) ")" )))))) ; theorem :: HALLMAR1:25 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Var "A")) "is" ($#m1_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "," (Set (Var "i")))))) ; theorem :: HALLMAR1:26 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "holds" (Bool (Set (Var "A")) "is" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A"))))) ; definitionlet "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); assume (Bool (Set (Const "A")) "is" ($#v2_relat_1 :::"non-empty"::: ) ) ; mode :::"Singlification"::: "of" "A" -> ($#m2_hallmar1 :::"Reduction"::: ) "of" "A" means :: HALLMAR1:def 8 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "A"))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" it ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))); end; :: deftheorem defines :::"Singlification"::: HALLMAR1:def 8 : (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_relat_1 :::"non-empty"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m4_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ")" )))); theorem :: HALLMAR1:27 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_relat_1 :::"non-empty"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m4_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A"))) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "f")) "is" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "i"))) ")" ) ")" ) ")" )))) ; registrationlet "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); let "k" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m3_hallmar1 :::"Singlification"::: ) "of" "A" "," "k"; end; registrationlet "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "F"))); cluster -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m4_hallmar1 :::"Singlification"::: ) "of" "A"; end; begin theorem :: HALLMAR1:28 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "B")) "being" ($#m2_hallmar1 :::"Reduction"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "X")) ($#r1_hallmar1 :::"is_a_system_of_different_representatives_of"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "X")) ($#r1_hallmar1 :::"is_a_system_of_different_representatives_of"::: ) (Set (Var "A"))))))) ; theorem :: HALLMAR1:29 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) )) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set ($#k2_hallmar1 :::"Cut"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ) "is" ($#v1_hallmar1 :::"Hall"::: ) ) ")" ))))) ; theorem :: HALLMAR1:30 (Bool "for" (Set (Var "F")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A")))) & (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) )) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m3_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "," (Set (Var "i")) "st" (Bool (Set (Var "G")) "is" ($#v1_hallmar1 :::"Hall"::: ) ))))) ; theorem :: HALLMAR1:31 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) )) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m4_hallmar1 :::"Singlification"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Var "G")) "is" ($#v1_hallmar1 :::"Hall"::: ) )))) ; theorem :: HALLMAR1:32 (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "F"))) "holds" (Bool "(" (Bool "ex" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "X")) ($#r1_hallmar1 :::"is_a_system_of_different_representatives_of"::: ) (Set (Var "A")))) "iff" (Bool (Set (Var "A")) "is" ($#v1_hallmar1 :::"Hall"::: ) ) ")" ))) ;