:: FINSEQ_4 semantic presentation begin definitionlet "f" be ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_one-to-one_at"::: "x" means :: FINSEQ_4:def 1 (Bool (Set "f" ($#k8_relat_1 :::"""::: ) (Set "(" ($#k9_relat_1 :::"Im"::: ) "(" "f" "," "x" ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) "x" ($#k1_tarski :::"}"::: ) )); end; :: deftheorem defines :::"is_one-to-one_at"::: FINSEQ_4:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x"))) "iff" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set "(" ($#k9_relat_1 :::"Im"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ))); theorem :: FINSEQ_4:1 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_4:2 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#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_4:3 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "z")))) ")" ) ")" ) ")" ))) ; theorem :: FINSEQ_4:4 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x"))) ")" ) "iff" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; definitionlet "R" be ($#m1_hidden :::"Relation":::); let "y" be ($#m1_hidden :::"set"::: ) ; pred "R" :::"just_once_values"::: "y" means :: FINSEQ_4:def 2 (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k10_relat_1 :::"Coim"::: ) "(" "R" "," "y" ")" ")" )) ($#r1_hidden :::"="::: ) (Num 1)); end; :: deftheorem defines :::"just_once_values"::: FINSEQ_4:def 2 : (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "R")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y"))) "iff" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k10_relat_1 :::"Coim"::: ) "(" (Set (Var "R")) "," (Set (Var "y")) ")" ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" ))); theorem :: FINSEQ_4:5 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_4:6 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y"))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )))) ")" ))) ; theorem :: FINSEQ_4:7 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y"))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" ) ")" )) ")" ))) ; theorem :: FINSEQ_4:8 (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 "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y")))) ")" )) ; theorem :: FINSEQ_4:9 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" ))) ; definitionlet "f" be ($#m1_hidden :::"Function":::); let "y" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Const "y"))) ; func "f" :::"<-"::: "y" -> ($#m1_hidden :::"set"::: ) means :: FINSEQ_4:def 3 (Bool "(" (Bool it ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) "y") ")" ); end; :: deftheorem defines :::"<-"::: FINSEQ_4:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_finseq_4 :::"<-"::: ) (Set (Var "y")))) "iff" (Bool "(" (Bool (Set (Var "b3")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ) ")" )))); theorem :: FINSEQ_4:10 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k9_relat_1 :::"Im"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "f")) ($#k1_finseq_4 :::"<-"::: ) (Set (Var "y")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FINSEQ_4:11 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "f")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_4 :::"<-"::: ) (Set (Var "y")) ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FINSEQ_4:12 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_finseq_4 :::"<-"::: ) (Set (Var "y")))))) ; theorem :: FINSEQ_4:13 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_finseq_4 :::"<-"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: FINSEQ_4:14 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "f")) ($#r1_finseq_4 :::"is_one-to-one_at"::: ) (Set (Set (Var "f")) ($#k1_finseq_4 :::"<-"::: ) (Set (Var "y")))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "d1", "d2" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"d1" "," "d2":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "d1", "d2", "d3" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"d1" "," "d2" "," "d3":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: FINSEQ_4:15 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "P")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P"))))) "holds" (Bool (Set (Set (Var "P")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_4:16 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "d")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "d"))))) ; theorem :: FINSEQ_4:17 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d1")) "," (Set (Var "d2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) ($#k2_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "d1"))) & (Bool (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) ($#k2_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "d2"))) ")" ))) ; theorem :: FINSEQ_4:18 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d1")) "," (Set (Var "d2")) "," (Set (Var "d3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) "," (Set (Var "d3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "d1"))) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) "," (Set (Var "d3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "d2"))) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d1")) "," (Set (Var "d2")) "," (Set (Var "d3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "d3"))) ")" ))) ; definitionlet "p" be ($#m1_hidden :::"FinSequence":::); let "x" be ($#m1_hidden :::"set"::: ) ; func "x" :::".."::: "p" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: FINSEQ_4:def 4 (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" "p" ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) "x" ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)); end; :: deftheorem defines :::".."::: FINSEQ_4:def 4 : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1))))); theorem :: FINSEQ_4:19 (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 (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: FINSEQ_4:20 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_4:21 (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 "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")))) & (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) ")" ))) ; theorem :: FINSEQ_4:22 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) & (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" )) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" ))) ; theorem :: FINSEQ_4:23 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: FINSEQ_4:24 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))))) ; theorem :: FINSEQ_4:25 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_4 :::"<-"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_4:26 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))))) ; theorem :: FINSEQ_4:27 (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")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"<>"::: ) (Set (Var "x"))) ")" )) "holds" (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))))) ; theorem :: FINSEQ_4:28 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k1_tarski :::"}"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ")" ) ")" ))) ; theorem :: FINSEQ_4:29 (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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k1_tarski :::"}"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: FINSEQ_4:30 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))) "iff" (Bool (Set ($#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_4:31 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))))) "implies" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" & "(" (Bool (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "implies" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ")" )))) ; theorem :: FINSEQ_4:32 (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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))))) "implies" (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")))) ")" & "(" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))))) "implies" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "implies" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) ")" & "(" (Bool (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "implies" (Bool (Set (Set "(" (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ")" )))) ; definitionlet "p" be ($#m1_hidden :::"FinSequence":::); let "x" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Const "p")))) ; func "p" :::"-|"::: "x" -> ($#m1_hidden :::"FinSequence":::) means :: FINSEQ_4:def 5 (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" "x" ($#k4_finseq_4 :::".."::: ) "p" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) & (Bool it ($#r1_hidden :::"="::: ) (Set "p" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ))) ")" )); end; :: deftheorem defines :::"-|"::: FINSEQ_4:def 5 : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")))) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) & (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ))) ")" )) ")" )))); theorem :: FINSEQ_4:33 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x"))))))) ; theorem :: FINSEQ_4:34 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))))) ; theorem :: FINSEQ_4:35 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1)))) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))))) ; theorem :: FINSEQ_4:36 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))))))) ; theorem :: FINSEQ_4:37 (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 "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )))))) ; theorem :: FINSEQ_4:38 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FINSEQ_4:39 (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 (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_4:40 (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 "(" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; theorem :: FINSEQ_4:41 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))))) ; definitionlet "p" be ($#m1_hidden :::"FinSequence":::); let "x" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Const "p")))) ; func "p" :::"|--"::: "x" -> ($#m1_hidden :::"FinSequence":::) means :: FINSEQ_4:def 6 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) "p" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" "x" ($#k4_finseq_4 :::".."::: ) "p" ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "p" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Set "(" "x" ($#k4_finseq_4 :::".."::: ) "p" ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"|--"::: FINSEQ_4:def 6 : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" ) ")" ))) ")" ) ")" ) ")" )))); theorem :: FINSEQ_4:42 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))))) ; theorem :: FINSEQ_4:43 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )))) "holds" (Bool (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_4:44 (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 (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_4:45 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )))) ")" ) ")" ))) ; theorem :: FINSEQ_4:46 (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")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )))))) ; theorem :: FINSEQ_4:47 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ) ")" ))) ; theorem :: FINSEQ_4:48 (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")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FINSEQ_4:49 (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 "(" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) "iff" (Bool (Set (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; theorem :: FINSEQ_4:50 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))))) ; theorem :: FINSEQ_4:51 (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 (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" ))))) ; theorem :: FINSEQ_4:52 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: FINSEQ_4:53 (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")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: FINSEQ_4:54 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" ))) ")" ) ")" ))) ; theorem :: FINSEQ_4:55 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" ))))) ; theorem :: FINSEQ_4:56 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set (Var "p")) ($#k1_finseq_3 :::"-"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" )))) "holds" (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: FINSEQ_4:57 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")) ")" ))))) ; theorem :: FINSEQ_4:58 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ))) ; theorem :: FINSEQ_4:59 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) ; theorem :: FINSEQ_4:60 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FINSEQ_4:61 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "q")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")))) & (Bool (Set (Var "q")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) ; theorem :: FINSEQ_4:62 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) ")" )) ; theorem :: FINSEQ_4:63 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "B")))) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) ; theorem :: FINSEQ_4:64 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "B")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: FINSEQ_4:65 (Bool "for" (Set (Var "B")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "B"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A")))) & (Bool (Set (Var "B")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "y")))) ")" )))) ; theorem :: FINSEQ_4:66 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "B")) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "B"))))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"<>"::: ) (Set (Var "x"))) ")" ) ")" )))) ; begin theorem :: FINSEQ_4:67 (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" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_4:68 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "E")) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))))))) ; theorem :: FINSEQ_4:69 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "E")) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))))))) ; theorem :: FINSEQ_4:70 (Bool "for" (Set (Var "a")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "m")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "m")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k7_partfun1 :::"/."::: ) (Set (Var "a"))))))) ; theorem :: FINSEQ_4:71 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "(" (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "m")))) ")" )))) ; theorem :: FINSEQ_4:72 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "X"))))) "holds" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))))) ; theorem :: FINSEQ_4:73 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k10_relat_1 :::"Coim"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" )) ($#r1_hidden :::"="::: ) (Num 1)))) ; definitionlet "x1", "x2", "x3", "x4" be ($#m1_hidden :::"set"::: ) ; func :::"<*":::"x1" "," "x2" "," "x3" "," "x4":::"*>"::: -> ($#m1_hidden :::"set"::: ) equals :: FINSEQ_4:def 7 (Set (Set ($#k11_finseq_1 :::"<*"::: ) "x1" "," "x2" "," "x3" ($#k11_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) "x4" ($#k9_finseq_1 :::"*>"::: ) )); let "x5" be ($#m1_hidden :::"set"::: ) ; func :::"<*":::"x1" "," "x2" "," "x3" "," "x4" "," "x5":::"*>"::: -> ($#m1_hidden :::"set"::: ) equals :: FINSEQ_4:def 8 (Set (Set ($#k11_finseq_1 :::"<*"::: ) "x1" "," "x2" "," "x3" ($#k11_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "x4" "," "x5" ($#k10_finseq_1 :::"*>"::: ) )); end; :: deftheorem defines :::"<*"::: FINSEQ_4:def 7 : (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) )))); :: deftheorem defines :::"<*"::: FINSEQ_4:def 8 : (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x4")) "," (Set (Var "x5")) ($#k10_finseq_1 :::"*>"::: ) )))); registrationlet "x1", "x2", "x3", "x4" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_finseq_4 :::"<*"::: ) "x1" "," "x2" "," "x3" "," "x4" ($#k7_finseq_4 :::"*>"::: ) ) -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; let "x5" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k8_finseq_4 :::"<*"::: ) "x1" "," "x2" "," "x3" "," "x4" "," "x5" ($#k8_finseq_4 :::"*>"::: ) ) -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "x1", "x2", "x3", "x4" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_finseq_4 :::"<*"::: ) "x1" "," "x2" "," "x3" "," "x4" ($#k7_finseq_4 :::"*>"::: ) ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; let "x5" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k8_finseq_4 :::"<*"::: ) "x1" "," "x2" "," "x3" "," "x4" "," "x5" ($#k8_finseq_4 :::"*>"::: ) ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x1", "x2", "x3", "x4" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"x1" "," "x2" "," "x3" "," "x4":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x1", "x2", "x3", "x4", "x5" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"x1" "," "x2" "," "x3" "," "x4" "," "x5":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: FINSEQ_4:74 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x3")) "," (Set (Var "x4")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ))) ")" )) ; theorem :: FINSEQ_4:75 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x4")) "," (Set (Var "x5")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k7_finseq_4 :::"*>"::: ) ))) ")" )) ; theorem :: FINSEQ_4:76 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k7_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) ($#k7_finseq_4 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 4)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) ")" ) ")" ))) ; theorem :: FINSEQ_4:77 canceled; theorem :: FINSEQ_4:78 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k8_finseq_4 :::"<*"::: ) (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) ($#k8_finseq_4 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 5)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "x5"))) ")" ) ")" ))) ; theorem :: FINSEQ_4:79 canceled; theorem :: FINSEQ_4:80 (Bool "for" (Set (Var "ND")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "ND")) "holds" (Bool "(" (Bool (Set (Set ($#k9_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) ($#k9_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "y1"))) & (Bool (Set (Set ($#k9_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) ($#k9_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "y2"))) & (Bool (Set (Set ($#k9_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) ($#k9_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "y3"))) & (Bool (Set (Set ($#k9_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) ($#k9_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "y4"))) ")" ))) ; theorem :: FINSEQ_4:81 (Bool "for" (Set (Var "ND")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "," (Set (Var "y5")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "ND")) "holds" (Bool "(" (Bool (Set (Set ($#k10_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "," (Set (Var "y5")) ($#k10_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "y1"))) & (Bool (Set (Set ($#k10_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "," (Set (Var "y5")) ($#k10_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "y2"))) & (Bool (Set (Set ($#k10_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "," (Set (Var "y5")) ($#k10_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "y3"))) & (Bool (Set (Set ($#k10_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "," (Set (Var "y5")) ($#k10_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "y4"))) & (Bool (Set (Set ($#k10_finseq_4 :::"<*"::: ) (Set (Var "y1")) "," (Set (Var "y2")) "," (Set (Var "y3")) "," (Set (Var "y4")) "," (Set (Var "y5")) ($#k10_finseq_4 :::"*>"::: ) ) ($#k7_partfun1 :::"/."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "y5"))) ")" ))) ; scheme :: FINSEQ_4:sch 1 Sch1{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" )))) "holds" (Bool P1[(Set (Var "n")) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")))]) ")" ) ")" )) provided (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" )))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool P1[(Set (Var "n")) "," (Set (Var "d"))]))) proof end; theorem :: FINSEQ_4:82 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r1_tarski :::"c="::: ) (Set (Var "q")))) "holds" (Bool "ex" (Set (Var "p9")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Set (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "p9"))) ($#r1_hidden :::"="::: ) (Set (Var "q")))))) ; theorem :: FINSEQ_4:83 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_tarski :::"c="::: ) (Set (Var "q"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; scheme :: FINSEQ_4:sch 2 PiLambdaD{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"Nat":::), F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "n")) ")" )) ")" ) ")" )) proof end; registrationlet "x1", "x2", "x3", "x4" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_finseq_4 :::"<*"::: ) "x1" "," "x2" "," "x3" "," "x4" ($#k7_finseq_4 :::"*>"::: ) ) -> (Num 4) ($#v3_card_1 :::"-element"::: ) ; let "x5" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k8_finseq_4 :::"<*"::: ) "x1" "," "x2" "," "x3" "," "x4" "," "x5" ($#k8_finseq_4 :::"*>"::: ) ) -> (Num 5) ($#v3_card_1 :::"-element"::: ) ; end; begin theorem :: FINSEQ_4:84 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Set (Var "p")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) ")" ))) ; theorem :: FINSEQ_4:85 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "D1")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "D1")) "," (Set (Var "D2")) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v3_funct_2 :::"bijective"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v3_funct_2 :::"bijective"::: ) )) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) "is" ($#v3_funct_2 :::"bijective"::: ) ))))) ; theorem :: FINSEQ_4:86 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "E1")) "," (Set (Var "E2")) "being" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set ($#k8_eqrel_1 :::"Class"::: ) (Set (Var "E1"))) ($#r1_hidden :::"="::: ) (Set ($#k8_eqrel_1 :::"Class"::: ) (Set (Var "E2"))))) "holds" (Bool (Set (Var "E1")) ($#r1_hidden :::"="::: ) (Set (Var "E2"))))) ; registrationlet "Z" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v1_finset_1 :::"finite"::: ) for ($#m1_eqrel_1 :::"a_partition"::: ) "of" "Z"; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) bbbadV5_FINSET_1() ($#v1_setfam_1 :::"with_non-empty_elements"::: ) for ($#m1_eqrel_1 :::"a_partition"::: ) "of" "X"; end; theorem :: FINSEQ_4:87 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "PX")) "being" ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "Pi")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Pi")) ($#r2_hidden :::"in"::: ) (Set (Var "PX")))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Pi"))))))) ; theorem :: FINSEQ_4:88 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "PX")) "being" ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "PX"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "X")))))) ; theorem :: FINSEQ_4:89 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "PA1")) "," (Set (Var "PA2")) "being" ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "PA1")) ($#r1_setfam_1 :::"is_finer_than"::: ) (Set (Var "PA2")))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "PA2"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "PA1")))))) ; theorem :: FINSEQ_4:90 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "PA1")) "," (Set (Var "PA2")) "being" ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "PA1")) ($#r1_setfam_1 :::"is_finer_than"::: ) (Set (Var "PA2")))) "holds" (Bool "for" (Set (Var "p2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "PA2")) (Bool "ex" (Set (Var "p1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "PA1")) "st" (Bool (Set (Var "p1")) ($#r1_tarski :::"c="::: ) (Set (Var "p2"))))))) ; theorem :: FINSEQ_4:91 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "PA1")) "," (Set (Var "PA2")) "being" ($#m1_eqrel_1 :::"a_partition"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set (Var "PA1")) ($#r1_setfam_1 :::"is_finer_than"::: ) (Set (Var "PA2"))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "PA1"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "PA2"))))) "holds" (Bool (Set (Var "PA1")) ($#r1_hidden :::"="::: ) (Set (Var "PA2"))))) ; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Const "D")) ($#k13_finseq_1 :::"*"::: ) ); let "k" be ($#m1_hidden :::"Nat":::); cluster (Set "M" ($#k7_partfun1 :::"/."::: ) "k") -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ; end; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Const "D")) ($#k13_finseq_1 :::"*"::: ) ); let "k" be ($#m1_hidden :::"Nat":::); cluster (Set "M" ($#k7_partfun1 :::"/."::: ) "k") -> "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_hidden :::"Function":::); end;