:: FINSEQ_5 semantic presentation begin theorem :: FINSEQ_5:1 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ))) ; theorem :: FINSEQ_5:2 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_5:3 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ))) ; theorem :: FINSEQ_5:4 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ")" )) ; theorem :: FINSEQ_5:5 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "y1")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y1")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y2")) ($#k1_tarski :::"}"::: ) )))) "holds" (Bool (Set (Var "y1")) ($#r1_hidden :::"="::: ) (Set (Var "y2"))))) ; registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> ($#v1_zfmisc_1 :::"trivial"::: ) ; let "y" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FINSEQ_5:6 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) ")" )) ; theorem :: FINSEQ_5:7 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::) (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_5:8 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FINSEQ_5:9 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ; theorem :: FINSEQ_5:10 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) ))))) ; theorem :: FINSEQ_5:11 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "i"))))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: FINSEQ_5:12 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")))) ")" )) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: FINSEQ_5:13 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")))) ")" )) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: FINSEQ_5:14 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_5:15 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_5:16 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_5:17 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))))) ; theorem :: FINSEQ_5:18 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_5:19 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_5:20 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k9_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_5:21 (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 "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; registrationlet "i" be ($#m1_hidden :::"Nat":::); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); cluster (Set "f" ($#k16_finseq_1 :::"|"::: ) "i") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: FINSEQ_5:22 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_5:23 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; theorem :: FINSEQ_5:24 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ))))))) ; theorem :: FINSEQ_5:25 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_5:26 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_5:27 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "i")) ")" )))))) ; theorem :: FINSEQ_5:28 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; theorem :: FINSEQ_5:29 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ))))) ; theorem :: FINSEQ_5:30 (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 "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: FINSEQ_5:31 (Bool "for" (Set (Var "j")) "," (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 "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "j")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "j"))))))) ; theorem :: FINSEQ_5:32 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i"))) "is" ($#v1_xboole_0 :::"empty"::: ) )))) ; theorem :: FINSEQ_5:33 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))))) ; registrationlet "i" be ($#m1_hidden :::"Nat":::); let "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); cluster (Set "f" ($#k1_rfinseq :::"/^"::: ) "i") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: FINSEQ_5:34 (Bool "for" (Set (Var "n")) "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 "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" )))))) ; theorem :: FINSEQ_5:35 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" )))))) ; theorem :: FINSEQ_5:36 (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 "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_5:37 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: FINSEQ_5:38 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_5:39 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))))) ; theorem :: FINSEQ_5:40 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")))))))) ; theorem :: FINSEQ_5:41 (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 "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "i")))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_hidden :::"set"::: ) ; func "f" :::"-:"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" equals :: FINSEQ_5:def 1 (Set "f" ($#k17_finseq_1 :::"|"::: ) (Set "(" "p" ($#k4_finseq_4 :::".."::: ) "f" ")" )); end; :: deftheorem defines :::"-:"::: FINSEQ_5:def 1 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" )))))); theorem :: FINSEQ_5:42 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_5:43 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")))))))) ; theorem :: FINSEQ_5:44 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)))))) ; theorem :: FINSEQ_5:45 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_5:46 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ))))))) ; theorem :: FINSEQ_5:47 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))))) ; theorem :: FINSEQ_5:48 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); let "f" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); cluster (Set "f" ($#k1_finseq_5 :::"-:"::: ) "p") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); func "f" :::":-"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" equals :: FINSEQ_5:def 2 (Set (Set ($#k12_finseq_1 :::"<*"::: ) "p" ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" "f" ($#k2_rfinseq :::"/^"::: ) (Set "(" "p" ($#k4_finseq_4 :::".."::: ) "f" ")" ) ")" )); end; :: deftheorem defines :::":-"::: FINSEQ_5:def 2 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" )))))); theorem :: FINSEQ_5:49 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i")))) ")" ))))) ; theorem :: FINSEQ_5:50 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1)))))) ; theorem :: FINSEQ_5:51 (Bool "for" (Set (Var "j")) "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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "j")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Set (Var "j")) ($#k1_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))))))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); cluster (Set "f" ($#k2_finseq_5 :::":-"::: ) "p") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: FINSEQ_5:52 (Bool "for" (Set (Var "j")) "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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "j")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "j")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "j")) ($#k1_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: FINSEQ_5:53 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_5:54 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: FINSEQ_5:55 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_5:56 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) )))) ; definitionlet "f" be ($#m1_hidden :::"FinSequence":::); func :::"Rev"::: "f" -> ($#m1_hidden :::"FinSequence":::) means :: FINSEQ_5:def 3 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) "f" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ); involutiveness (Bool "for" (Set (Var "b1")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b1"))))) "holds" (Bool (Set (Set (Var "b1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))) ")" )) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b1")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b2"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "b1")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "b1")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" )) ; end; :: deftheorem defines :::"Rev"::: FINSEQ_5:def 3 : (Bool "for" (Set (Var "f")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b2"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) ")" )); theorem :: FINSEQ_5:57 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ))) ")" )) ; theorem :: FINSEQ_5:58 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: FINSEQ_5:59 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ))))) ; registrationlet "f" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set ($#k3_finseq_5 :::"Rev"::: ) "f") -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: FINSEQ_5:60 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_5:61 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x2")) "," (Set (Var "x1")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_5:62 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))) ")" )) ; registrationlet "f" be ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set ($#k3_finseq_5 :::"Rev"::: ) "f") -> ($#v2_funct_1 :::"one-to-one"::: ) ; end; theorem :: FINSEQ_5:63 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ))))) ; theorem :: FINSEQ_5:64 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "g")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" )))) ; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); :: original: :::"Rev"::: redefine func :::"Rev"::: "f" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: FINSEQ_5:65 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1))) ")" ))) ; theorem :: FINSEQ_5:66 (Bool "for" (Set (Var "j")) "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 "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")))))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); let "n" be ($#m1_hidden :::"Nat":::); func :::"Ins"::: "(" "f" "," "n" "," "p" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" equals :: FINSEQ_5:def 4 (Set (Set "(" (Set "(" "f" ($#k17_finseq_1 :::"|"::: ) "n" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) "p" ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" "f" ($#k2_rfinseq :::"/^"::: ) "n" ")" )); end; :: deftheorem defines :::"Ins"::: FINSEQ_5:def 4 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" ))))))); theorem :: FINSEQ_5:67 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_5:68 (Bool "for" (Set (Var "n")) "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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: FINSEQ_5:69 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))))))) ; theorem :: FINSEQ_5:70 (Bool "for" (Set (Var "n")) "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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ))))))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "n" be ($#m1_hidden :::"Nat":::); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k5_finseq_5 :::"Ins"::: ) "(" "f" "," "n" "," "p" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: FINSEQ_5:71 (Bool "for" (Set (Var "n")) "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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" ))))))) ; theorem :: FINSEQ_5:72 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))))))))) ; theorem :: FINSEQ_5:73 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_5:74 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))))))))) ; theorem :: FINSEQ_5:75 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool (Set (Set "(" ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1))))))) ; theorem :: FINSEQ_5:76 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) "holds" (Bool (Set ($#k5_finseq_5 :::"Ins"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "p")) ")" ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))))) ; begin theorem :: FINSEQ_5:77 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i1")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i1")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i2")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i1")))) ")" )))) ; theorem :: FINSEQ_5:78 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D")))))) ; theorem :: FINSEQ_5:79 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D"))))) ; registration cluster ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FINSEQ_5:80 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "l1")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "l2")) ($#k7_nat_d :::"-'"::: ) (Set (Var "l1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "l2")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "l1"))))))) ; theorem :: FINSEQ_5:81 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2) ")" ) ($#k10_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_5:82 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "k")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )))))) ; theorem :: FINSEQ_5:83 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )))))) ; theorem :: FINSEQ_5:84 (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")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "p")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" )))))) ;