:: FINSEQ_3 semantic presentation begin theorem :: FINSEQ_3:1 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k1_enumset1 :::"}"::: ) )) ; theorem :: FINSEQ_3:2 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set ($#k2_enumset1 :::"{"::: ) (Num 1) "," (Num 2) "," (Num 3) "," (Num 4) ($#k2_enumset1 :::"}"::: ) )) ; theorem :: FINSEQ_3:3 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set ($#k3_enumset1 :::"{"::: ) (Num 1) "," (Num 2) "," (Num 3) "," (Num 4) "," (Num 5) ($#k3_enumset1 :::"}"::: ) )) ; theorem :: FINSEQ_3:4 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set ($#k4_enumset1 :::"{"::: ) (Num 1) "," (Num 2) "," (Num 3) "," (Num 4) "," (Num 5) "," (Num 6) ($#k4_enumset1 :::"}"::: ) )) ; theorem :: FINSEQ_3:5 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 7)) ($#r1_hidden :::"="::: ) (Set ($#k5_enumset1 :::"{"::: ) (Num 1) "," (Num 2) "," (Num 3) "," (Num 4) "," (Num 5) "," (Num 6) "," (Num 7) ($#k5_enumset1 :::"}"::: ) )) ; theorem :: FINSEQ_3:6 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Set ($#k6_enumset1 :::"{"::: ) (Num 1) "," (Num 2) "," (Num 3) "," (Num 4) "," (Num 5) "," (Num 6) "," (Num 7) "," (Num 8) ($#k6_enumset1 :::"}"::: ) )) ; theorem :: FINSEQ_3:7 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Bool "not" (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) ")" )) ; theorem :: FINSEQ_3:8 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Bool "not" (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) ; theorem :: FINSEQ_3:9 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n")) ")" )))) ; theorem :: FINSEQ_3:10 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "holds" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: FINSEQ_3:11 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_3:12 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_3:13 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))) "iff" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) ")" )) ; theorem :: FINSEQ_3:14 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: FINSEQ_3:15 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: FINSEQ_3:16 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_hidden :::"<>"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FINSEQ_3:17 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: FINSEQ_3:18 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n")) ")" )))) ; theorem :: FINSEQ_3:19 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) "," (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))) ($#r3_xboole_0 :::"are_c=-comparable"::: ) )) ; theorem :: FINSEQ_3:20 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Num 1)) ")" ))) ; theorem :: FINSEQ_3:21 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Num 1) "," (Num 2) ($#k2_tarski :::"}"::: ) )) ")" ))) ; theorem :: FINSEQ_3:22 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ))))) ; theorem :: FINSEQ_3:23 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )))) ; theorem :: FINSEQ_3:24 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: FINSEQ_3:25 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) ")" ) ")" ))) ; theorem :: FINSEQ_3:26 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))) "iff" (Bool "(" (Bool (Set (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) & (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "n"))) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" ) ")" ))) ; theorem :: FINSEQ_3:27 canceled; theorem :: FINSEQ_3:28 canceled; theorem :: FINSEQ_3:29 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")))) "iff" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "q")))) ")" )) ; theorem :: FINSEQ_3:30 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")))) "iff" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "q")))) ")" )) ; theorem :: FINSEQ_3:31 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:32 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) ; theorem :: FINSEQ_3:33 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:34 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:35 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:36 (Bool "for" (Set (Var "u")) "," (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "u")) ($#k9_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:37 (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "u")) "," (Set (Var "v")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:38 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" )) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))))) ; theorem :: FINSEQ_3:39 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A")))))) ; theorem :: FINSEQ_3:40 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FINSEQ_3:41 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "l")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "i")))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "l"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Var "k"))) & (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "l")))) "holds" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_3:42 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "i")))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "j"))))) "holds" (Bool "(" (Bool "(" "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) ")" ) "iff" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "Y")) ")" ))) ")" ))) ; theorem :: FINSEQ_3:43 (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; theorem :: FINSEQ_3:44 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "n")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "n")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:45 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "n")) "," (Set (Var "m")) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "n")) "," (Set (Var "m")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:46 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "k")))) ; theorem :: FINSEQ_3:47 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) ; theorem :: FINSEQ_3:48 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "k"))))) ; theorem :: FINSEQ_3:49 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) "iff" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" ))) ; theorem :: FINSEQ_3:50 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "k")) ")" ) ")" ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_3:51 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "m")))) "iff" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" )) ; theorem :: FINSEQ_3:52 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")))) "iff" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) ")" )) ; theorem :: FINSEQ_3:53 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "l")))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "k"))))) ; theorem :: FINSEQ_3:54 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "l")))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) ; theorem :: FINSEQ_3:55 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) ))))) ; theorem :: FINSEQ_3:56 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set (Var "X"))) "is" ($#m1_hidden :::"FinSequence":::)) "iff" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) ")" ))) ; theorem :: FINSEQ_3:57 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "q")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: FINSEQ_3:58 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "A")))))) ; definitionlet "p" be ($#m1_hidden :::"FinSequence":::); let "A" be ($#m1_hidden :::"set"::: ) ; func "p" :::"-"::: "A" -> ($#m1_hidden :::"FinSequence":::) equals :: FINSEQ_3:def 1 (Set "p" ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) "p" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" "p" ($#k8_relat_1 :::"""::: ) "A" ")" ) ")" ) ")" )); end; :: deftheorem defines :::"-"::: FINSEQ_3:def 1 : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ")" ) ")" ))))); theorem :: FINSEQ_3:59 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ")" ))))) ; theorem :: FINSEQ_3:60 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:61 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:62 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ")" )))) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))))) ; theorem :: FINSEQ_3:63 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:64 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:65 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set (Var "A")))))) ; theorem :: FINSEQ_3:66 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:67 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:68 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ))) ; theorem :: FINSEQ_3:69 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) "iff" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) ")" ))) ; theorem :: FINSEQ_3:70 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) "iff" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) ")" ))) ; theorem :: FINSEQ_3:71 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p")))) ; theorem :: FINSEQ_3:72 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FINSEQ_3:73 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "q")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FINSEQ_3:74 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FINSEQ_3:75 (Bool "for" (Set (Var "x")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) "iff" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ")" )) ; theorem :: FINSEQ_3:76 (Bool "for" (Set (Var "x")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" )) ; theorem :: FINSEQ_3:77 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" )) ; theorem :: FINSEQ_3:78 (Bool "for" (Set (Var "x")) "," (Set (Var "A")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:79 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ")" )) ; theorem :: FINSEQ_3:80 (Bool "for" (Set (Var "x")) "," (Set (Var "A")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:81 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" )) ; theorem :: FINSEQ_3:82 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ")" ) ")" )) ; theorem :: FINSEQ_3:83 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "iff" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")))) ")" )))) ; theorem :: FINSEQ_3:84 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")) ")" )))) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "iff" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "q")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) ))) ")" )))) ; theorem :: FINSEQ_3:85 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) "{" (Set (Var "k")) where k "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) "}" ) "or" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "or" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) ")" ))))) ; theorem :: FINSEQ_3:86 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "D")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))))) ; theorem :: FINSEQ_3:87 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: FINSEQ_3:88 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "A")) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")) ")" ) ")" ) ")" ))))) ; theorem :: FINSEQ_3:89 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "A")) ")" ))))) ; theorem :: FINSEQ_3:90 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))))) ; theorem :: FINSEQ_3:91 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "q")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "q")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ) "iff" (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FINSEQ_3:92 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "holds" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "A"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: FINSEQ_3:93 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FINSEQ_3:94 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) "iff" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FINSEQ_3:95 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x"))) ")" ) "iff" (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FINSEQ_3:96 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: FINSEQ_3:97 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_3:98 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 2)))) ; theorem :: FINSEQ_3:99 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k10_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_3:100 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) )) & (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 3)))) ; theorem :: FINSEQ_3:101 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_enumset1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k1_enumset1 :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 3)))) ; begin theorem :: FINSEQ_3:102 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "df")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Var "df")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D"))(Bool "ex" (Set (Var "df1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_hidden :::"="::: ) (Set (Set (Var "df")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "df")) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "d")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "df1")))) ")" ))))) ; theorem :: FINSEQ_3:103 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "df")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "d")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "df"))))) "holds" (Bool (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "d")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "df")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "df")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; definitionlet "i" be ($#m1_hidden :::"Nat":::); let "p" be ($#m1_hidden :::"FinSequence":::); func :::"Del"::: "(" "p" "," "i" ")" -> ($#m1_hidden :::"FinSequence":::) equals :: FINSEQ_3:def 2 (Set "p" ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) "p" ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) "i" ($#k1_tarski :::"}"::: ) ) ")" ) ")" )); end; :: deftheorem defines :::"Del"::: FINSEQ_3:def 2 : (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "p")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ))))); theorem :: FINSEQ_3:104 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))))) "implies" (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "p")) "," (Set (Var "i")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "m"))) ")" )) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))))) "implies" (Bool (Set ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "p")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ")" ))) ; theorem :: FINSEQ_3:105 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "p")) "," (Set (Var "i")) ")" ) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))))) ; theorem :: FINSEQ_3:106 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "p")) "," (Set (Var "i")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_3:107 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "m")))) ; theorem :: FINSEQ_3:108 (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool "(" "(" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "implies" (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "k")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "i"))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "implies" (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "k")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1))) ")" ")" )) ; theorem :: FINSEQ_3:109 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "m"))))) ; theorem :: FINSEQ_3:110 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))))) ; theorem :: FINSEQ_3:111 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" ($#k2_finseq_3 :::"Del"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: FINSEQ_3:112 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "n")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))))) ; theorem :: FINSEQ_3:113 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_tarski :::"c="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set (Var "q")) ($#k16_finseq_1 :::"|"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p")))) ; theorem :: FINSEQ_3:114 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set (Var "A")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set (Var "F")) ($#k8_relat_1 :::"""::: ) (Set "(" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set (Var "A")) ")" ) ")" ) ")" )) "is" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")) ")" )))) ; theorem :: FINSEQ_3:115 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")) ")" ) (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "F")) ")" ) "st" (Bool (Set (Set "(" (Set (Var "F")) ($#k1_finseq_3 :::"-"::: ) (Set "(" (Set (Var "A")) ($#k3_subset_1 :::"`"::: ) ")" ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "F")) ($#k1_finseq_3 :::"-"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k3_relat_1 :::"*"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_3:116 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSubsequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#m1_hidden :::"FinSequence":::))) "holds" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ; theorem :: FINSEQ_3:117 (Bool "for" (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool (Set (Var "t")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: FINSEQ_3:118 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")))) "iff" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p"))) ($#r2_xboole_0 :::"c<"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "q")))) ")" )) ; theorem :: FINSEQ_3:119 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) "iff" (Bool (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; registrationlet "i" be ($#m1_hidden :::"Nat":::); let "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set "i" ($#k4_finseq_2 :::"-tuples_on"::: ) "D") -> ($#v2_card_3 :::"with_common_domain"::: ) ; end; registrationlet "i" be ($#m1_hidden :::"Nat":::); let "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set "i" ($#k4_finseq_2 :::"-tuples_on"::: ) "D") -> ($#v3_card_3 :::"product-like"::: ) ; end; begin theorem :: FINSEQ_3:120 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D1")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "D1")) "," (Set (Var "D2")) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k4_finseqop :::"*"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k4_finseqop :::"*"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k4_finseqop :::"*"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k4_finseqop :::"*"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" )))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "D")); func :::"ExtendRel"::: "R" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" "D" ($#k3_finseq_2 :::"*"::: ) ")" ) means :: FINSEQ_3:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "x"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "y")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") ")" ) ")" ) ")" )); end; :: deftheorem defines :::"ExtendRel"::: FINSEQ_3:def 3 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_3 :::"ExtendRel"::: ) (Set (Var "R")))) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "x"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "y")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" ) ")" ) ")" )) ")" )))); theorem :: FINSEQ_3:121 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_finseq_3 :::"ExtendRel"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "D")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "(" (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ")" )))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set (Const "D")); let "y" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k8_eqrel_1 :::"Class"::: ) (Set (Const "R"))); let "x" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); pred "x" :::"is_representatives_FS"::: "y" means :: FINSEQ_3:def 4 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "x") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "y")) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "x"))) "holds" (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" "R" "," (Set "(" "x" ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set "y" ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) ")" ) ")" ); end; :: deftheorem defines :::"is_representatives_FS"::: FINSEQ_3:def 4 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k8_eqrel_1 :::"Class"::: ) (Set (Var "R"))) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_finseq_3 :::"is_representatives_FS"::: ) (Set (Var "y"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "x"))))) "holds" (Bool (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set (Var "R")) "," (Set "(" (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")))) ")" ) ")" ) ")" ))))); theorem :: FINSEQ_3:122 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "y")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k8_eqrel_1 :::"Class"::: ) (Set (Var "R"))) (Bool "ex" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Set (Var "x")) ($#r1_finseq_3 :::"is_representatives_FS"::: ) (Set (Var "y"))))))) ; theorem :: FINSEQ_3:123 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "X")) ($#k9_finseq_1 :::"*>"::: ) ))) "iff" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) )) ")" )) ")" )) ; theorem :: FINSEQ_3:124 (Bool "for" (Set (Var "z")) "," (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k10_finseq_1 :::"*>"::: ) ))) "iff" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ")" )) ")" )) ; theorem :: FINSEQ_3:125 (Bool "for" (Set (Var "a")) "," (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) ($#k11_finseq_1 :::"*>"::: ) ))) "iff" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) )) ")" )) ")" )) ; theorem :: FINSEQ_3:126 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "D")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "D"))))) ; theorem :: FINSEQ_3:127 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "D1")) "," (Set (Var "D2")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) ($#k10_finseq_1 :::"*>"::: ) ) where d1 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D1")), d2 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D2")) : (Bool verum) "}" )) ; theorem :: FINSEQ_3:128 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "D")) "," (Set (Var "D")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "D"))))) ; theorem :: FINSEQ_3:129 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "," (Set (Var "D3")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "D1")) "," (Set (Var "D2")) "," (Set (Var "D3")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) "," (Set (Var "d3")) ($#k11_finseq_1 :::"*>"::: ) ) where d1 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D1")), d2 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D2")), d3 "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D3")) : (Bool verum) "}" )) ; theorem :: FINSEQ_3:130 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "D")) "," (Set (Var "D")) "," (Set (Var "D")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Num 3) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "D"))))) ; theorem :: FINSEQ_3:131 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set "(" (Set (Var "i")) ($#k2_finseq_2 :::"|->"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "D")))))) ; theorem :: FINSEQ_3:132 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k2_funct_6 :::"doms"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "f")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) & (Bool (Set ($#k3_funct_6 :::"rngs"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "f")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ")" )) ; theorem :: FINSEQ_3:133 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k2_funct_6 :::"doms"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set ($#k3_funct_6 :::"rngs"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) ")" )) ; theorem :: FINSEQ_3:134 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k2_funct_6 :::"doms"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h")) ")" ) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set ($#k3_funct_6 :::"rngs"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h")) ")" ) ($#k11_finseq_1 :::"*>"::: ) )) ")" )) ; theorem :: FINSEQ_3:135 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "X")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set ($#k4_funct_6 :::"meet"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "X")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "X"))) ")" )) ; theorem :: FINSEQ_3:136 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")))) & (Bool (Set ($#k4_funct_6 :::"meet"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))) ")" )) ; theorem :: FINSEQ_3:137 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k3_card_3 :::"Union"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Z")))) & (Bool (Set ($#k4_funct_6 :::"meet"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Z")))) ")" )) ; theorem :: FINSEQ_3:138 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "f")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k5_funct_6 :::".."::: ) "(" (Num 1) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k5_funct_6 :::".."::: ) "(" (Num 1) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_funct_6 :::".."::: ) "(" (Num 1) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ))) ; theorem :: FINSEQ_3:139 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k5_funct_6 :::".."::: ) "(" (Num 2) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_funct_6 :::".."::: ) "(" (Num 2) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ))) ; theorem :: FINSEQ_3:140 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "h")) "," (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_funct_6 :::".."::: ) "(" (Num 3) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: FINSEQ_3:141 (Bool "for" (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k6_funct_6 :::"<:"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "h")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k6_funct_6 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set ($#k6_funct_6 :::"<:"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "h")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k6_funct_6 :::":>"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ")" ) ")" )) ; theorem :: FINSEQ_3:142 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k6_funct_6 :::"<:"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k6_funct_6 :::":>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2")) ")" )))) "holds" (Bool (Set (Set ($#k6_funct_6 :::"<:"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k6_funct_6 :::":>"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) ")" ) ")" )) ; theorem :: FINSEQ_3:143 (Bool "for" (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "h")) ($#k9_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h")) ")" ) ($#k9_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "h")) ($#k9_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h")) ")" ) ($#k9_finseq_1 :::"*>"::: ) ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "h")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_finseq_1 :::"*>"::: ) )) ")" ) ")" )) ; theorem :: FINSEQ_3:144 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f1")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2"))))) "holds" (Bool (Set (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) ")" ) ")" )) ; theorem :: FINSEQ_3:145 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2"))))) "holds" (Bool "for" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k13_funct_3 :::"<:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k13_funct_3 :::":>"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "y1")) "," (Set (Var "y2")) ($#k4_tarski :::"]"::: ) )) "iff" (Bool (Set (Set ($#k6_funct_6 :::"<:"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k6_funct_6 :::":>"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) ($#k10_finseq_1 :::"*>"::: ) )) ")" )))) ; theorem :: FINSEQ_3:146 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f2"))))) "holds" (Bool "for" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k15_funct_3 :::":]"::: ) ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "y1")) "," (Set (Var "y2")) ($#k4_tarski :::"]"::: ) )) "iff" (Bool (Set (Set "(" ($#k7_funct_6 :::"Frege"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "f1")) "," (Set (Var "f2")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) ($#k10_finseq_1 :::"*>"::: ) )) ")" )))) ; theorem :: FINSEQ_3:147 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k8_funct_6 :::"Funcs"::: ) "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "X")) ($#k9_finseq_1 :::"*>"::: ) ) "," (Set (Var "Y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:148 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k8_funct_6 :::"Funcs"::: ) "(" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k10_finseq_1 :::"*>"::: ) ) "," (Set (Var "Z")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "Z")) ")" ")" ) "," (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "Y")) "," (Set (Var "Z")) ")" ")" ) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:149 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_funct_6 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "Y")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:150 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_funct_6 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "Y")) "," (Set (Var "Z")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "Y")) ")" ")" ) "," (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set (Var "Z")) ")" ")" ) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_3:151 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ))) ; theorem :: FINSEQ_3:152 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) "holds" (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))))) "holds" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))))) ; registrationlet "i" be ($#m1_hidden :::"Nat":::); let "D" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "i" ($#k4_finseq_2 :::"-tuples_on"::: ) "D") -> ($#v1_finset_1 :::"finite"::: ) ; end;