:: FINSEQ_6 semantic presentation begin registrationlet "x", "y", "z" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k11_finseq_1 :::"<*"::: ) "x" "," "y" "," "z" ($#k11_finseq_1 :::"*>"::: ) ) -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ; end; registrationlet "f" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::); cluster (Set ($#k3_finseq_5 :::"Rev"::: ) "f") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; begin theorem :: FINSEQ_6:1 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "i")))) & (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: FINSEQ_6:2 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "k"))))) ; theorem :: FINSEQ_6:3 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p1")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:4 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p1")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:5 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:6 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f1")))))) ; theorem :: FINSEQ_6:7 (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f2")) ")" ))))) ; theorem :: FINSEQ_6:8 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")))))) ; theorem :: FINSEQ_6:9 (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_6:10 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Var "f1")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2"))))) ; theorem :: FINSEQ_6:11 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k5_relat_1 :::"|"::: ) (Set (Var "A")))))) ; theorem :: FINSEQ_6:12 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p")))))) ; registrationlet "f1" be ($#m1_hidden :::"FinSequence":::); let "i" be ($#m1_hidden :::"Nat":::); cluster (Set "f1" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "i" ")" )) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; theorem :: FINSEQ_6:13 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f1")) ($#r1_tarski :::"c="::: ) (Set (Var "f2")))) "holds" (Bool (Set (Set (Var "f3")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f3")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2"))))) ; theorem :: FINSEQ_6:14 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f2")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "i")) ")" ) ")" ))))) ; theorem :: FINSEQ_6:15 (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f2")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p")) ")" ))))) ; theorem :: FINSEQ_6:16 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2"))) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ))))) ; theorem :: FINSEQ_6:17 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2"))) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Var "f1")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p"))))) ; theorem :: FINSEQ_6:18 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: FINSEQ_6:19 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "p1")) ($#k4_finseq_4 :::".."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: FINSEQ_6:20 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set (Var "p2")) ($#k4_finseq_4 :::".."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 2))) ; theorem :: FINSEQ_6:21 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "p1")) ($#k4_finseq_4 :::".."::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: FINSEQ_6:22 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set (Var "p2")) ($#k4_finseq_4 :::".."::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 2))) ; theorem :: FINSEQ_6:23 (Bool "for" (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "p2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3")))) "holds" (Bool (Set (Set (Var "p3")) ($#k4_finseq_4 :::".."::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k11_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Num 3))) ; theorem :: FINSEQ_6:24 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) ))))) ; theorem :: FINSEQ_6:25 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ; theorem :: FINSEQ_6:26 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:27 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:28 (Bool "for" (Set (Var "x")) "," (Set (Var "z")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:29 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:30 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "z")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:31 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ))) ; theorem :: FINSEQ_6:32 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "z")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "z")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ; theorem :: FINSEQ_6:33 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FINSEQ_6:34 (Bool "for" (Set (Var "x")) "," (Set (Var "z")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool (Set (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FINSEQ_6:35 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "y")))))) ; theorem :: FINSEQ_6:36 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")))))) "holds" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))))) ; theorem :: FINSEQ_6:37 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))))) ; theorem :: FINSEQ_6:38 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x")))) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "x")))))) ; theorem :: FINSEQ_6:39 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x")))) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f"))) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "x"))))) ; begin theorem :: FINSEQ_6:40 (Bool "for" (Set (Var "D")) "being" ($#~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")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: FINSEQ_6:41 (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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p")) ")" )))))) ; theorem :: FINSEQ_6:42 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_6:43 (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")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: FINSEQ_6:44 (Bool "for" (Set (Var "D")) "being" ($#~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 "f")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "f"))) ")" )))) ; theorem :: FINSEQ_6:45 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "being" ($#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 "p1")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_6:46 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) ) ($#k2_rfinseq :::"/^"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p2")) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:47 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k2_rfinseq :::"/^"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p2")) "," (Set (Var "p3")) ($#k2_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:48 (Bool "for" (Set (Var "k")) "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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")))) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "k")))))) ; theorem :: FINSEQ_6:49 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:50 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:51 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3")))) "holds" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p3"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:52 (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")) "holds" (Bool "(" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ")" ))) ; theorem :: FINSEQ_6:53 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p2")) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:54 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p2")) "," (Set (Var "p3")) ($#k2_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:55 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p3")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3"))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3")))) "holds" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p3"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p3")) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:56 (Bool "for" (Set (Var "k")) "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 "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" ) ")" ))))))) ; theorem :: FINSEQ_6:57 (Bool "for" (Set (Var "k")) "being" ($#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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" ))))))) ; theorem :: FINSEQ_6:58 (Bool "for" (Set (Var "k")) "," (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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" )))))) ; theorem :: FINSEQ_6:59 (Bool "for" (Set (Var "k")) "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 "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")))))))) ; theorem :: FINSEQ_6:60 (Bool "for" (Set (Var "D")) "being" ($#~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 "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Num 1)))))) ; theorem :: FINSEQ_6:61 (Bool "for" (Set (Var "D")) "being" ($#~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 "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )))))) ; theorem :: FINSEQ_6:62 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "x")) ($#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")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ))))))) ; theorem :: FINSEQ_6:63 (Bool "for" (Set (Var "k")) "being" ($#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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ))))))) ; theorem :: FINSEQ_6:64 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2"))))))) ; theorem :: FINSEQ_6:65 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:66 (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 "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1"))))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:67 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f2")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f1")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f2")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" )))))) ; theorem :: FINSEQ_6:68 (Bool "for" (Set (Var "D")) "being" ($#~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")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:69 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p1")) ")" ) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p1")) ")" ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p2"))))))) ; theorem :: FINSEQ_6:70 (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 (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ")" )))))) ; theorem :: FINSEQ_6:71 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p1")) ")" ) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p1")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p2"))))))) ; theorem :: FINSEQ_6:72 (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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_6:73 (Bool "for" (Set (Var "i")) "being" ($#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 (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_6:74 (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")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:75 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "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 "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p1")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p1")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p2"))))))) ; theorem :: FINSEQ_6:76 (Bool "for" (Set (Var "D")) "being" ($#~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")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_6:77 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f2")))))) ; theorem :: FINSEQ_6:78 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "p2")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" )))))) ; theorem :: FINSEQ_6:79 (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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))))) ; theorem :: FINSEQ_6:80 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D")))))) ; theorem :: FINSEQ_6:81 (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set "(" (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k"))))))) ; theorem :: FINSEQ_6:82 (Bool "for" (Set (Var "k")) "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 "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")))))))) ; theorem :: FINSEQ_6:83 (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 ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:84 (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "being" ($#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")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_6:85 (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_6:86 (Bool "for" (Set (Var "D")) "being" ($#~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")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:87 (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" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p"))))))) ; theorem :: FINSEQ_6:88 (Bool "for" (Set (Var "D")) "being" ($#~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")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p"))))))) ; begin definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "IT" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); attr "IT" is :::"circular"::: means :: FINSEQ_6:def 1 (Bool (Set "IT" ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set "IT" ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "IT" ")" ))); end; :: deftheorem defines :::"circular"::: FINSEQ_6:def 1 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "IT")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_finseq_6 :::"circular"::: ) ) "iff" (Bool (Set (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "IT")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "IT")) ")" ))) ")" ))); definitionlet "D" be ($#~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 :::"Rotate"::: "(" "f" "," "p" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" equals :: FINSEQ_6:def 2 (Set (Set "(" "f" ($#k2_finseq_5 :::":-"::: ) "p" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set "(" "f" ($#k1_finseq_5 :::"-:"::: ) "p" ")" ) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" )) if (Bool "p" ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f")) otherwise "f"; end; :: deftheorem defines :::"Rotate"::: FINSEQ_6:def 2 : (Bool "for" (Set (Var "D")) "being" ($#~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 "(" "(" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "implies" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) "implies" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))) ")" ")" )))); registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() (Num 1) ($#v3_card_1 :::"-element"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_finseq_6 :::"circular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; cluster ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_finseq_6 :::"circular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: FINSEQ_6:89 (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 ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); let "f" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v1_finseq_6 :::"circular"::: ) ; end; theorem :: FINSEQ_6:90 (Bool "for" (Set (Var "D")) "being" ($#~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" ($#v1_finseq_6 :::"circular"::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_6:91 (Bool "for" (Set (Var "D")) "being" ($#~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 (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )))))) ; theorem :: FINSEQ_6:92 (Bool "for" (Set (Var "D")) "being" ($#~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 "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_6:93 (Bool "for" (Set (Var "D")) "being" ($#~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 ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ))))) ; theorem :: FINSEQ_6:94 (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")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:95 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) ) "," (Set (Var "p1")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:96 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) ) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p2")) "," (Set (Var "p2")) ($#k2_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:97 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set (Var "p1")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:98 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p2")) ($#k3_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:99 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p2")) "," (Set (Var "p3")) "," (Set (Var "p1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p2")) ($#r1_hidden :::"<>"::: ) (Set (Var "p3")))) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set (Var "p3")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "p3")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k3_finseq_4 :::"*>"::: ) )))) ; theorem :: FINSEQ_6:100 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_6:101 (Bool "for" (Set (Var "D")) "being" ($#~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 ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )))))) ; theorem :: FINSEQ_6:102 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p2")) "," (Set (Var "p1")) "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 "p2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set (Var "p1")) ")" ) ")" )))) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p1")) ")" ")" ) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p2")) ")" ))))) ; theorem :: FINSEQ_6:103 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p2")) "," (Set (Var "p1")) "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 (Set (Var "p2")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p1")) ")" ) ")" )))) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p1")) ")" ")" ) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p2")) ")" ))))) ; theorem :: FINSEQ_6:104 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p2")) "," (Set (Var "p1")) "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 "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ))) & (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p1")) ")" ")" ) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p2")) ")" ))))) ; theorem :: FINSEQ_6:105 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p2")) "," (Set (Var "p1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_finseq_6 :::"circular"::: ) ) & (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p1")) ")" ")" ) "," (Set (Var "p2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p2")) ")" ))))) ; theorem :: FINSEQ_6:106 (Bool "for" (Set (Var "D")) "being" ($#~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" ($#v1_finseq_6 :::"circular"::: ) ) & (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) "," (Set (Var "p")) ")" ))))) ; begin theorem :: FINSEQ_6:107 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v1_zfmisc_1 :::"trivial"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) ) "or" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k12_finseq_1 :::"*>"::: ) ))) ")" ))) ; begin theorem :: FINSEQ_6:108 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" ))) "or" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ))) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k5_real_1 :::"-"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ")" ))))) ; theorem :: FINSEQ_6:109 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) ")" )))) ; theorem :: FINSEQ_6:110 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ; theorem :: FINSEQ_6:111 (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 "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "k"))))))) ; theorem :: FINSEQ_6:112 (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")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "l1"))))))) ; definitionlet "f" be ($#m1_hidden :::"FinSequence":::); let "k1", "k2" be ($#m1_hidden :::"Nat":::); func :::"mid"::: "(" "f" "," "k1" "," "k2" ")" -> ($#m1_hidden :::"FinSequence":::) equals :: FINSEQ_6:def 3 (Set (Set "(" "f" ($#k1_rfinseq :::"/^"::: ) (Set "(" "k1" ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k16_finseq_1 :::"|"::: ) (Set "(" (Set "(" "k2" ($#k7_nat_d :::"-'"::: ) "k1" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) if (Bool "k1" ($#r1_xxreal_0 :::"<="::: ) "k2") otherwise (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set "(" "f" ($#k1_rfinseq :::"/^"::: ) (Set "(" "k2" ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k16_finseq_1 :::"|"::: ) (Set "(" (Set "(" "k1" ($#k7_nat_d :::"-'"::: ) "k2" ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )); end; :: deftheorem defines :::"mid"::: FINSEQ_6:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2")))) "implies" (Bool (Set ($#k2_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_rfinseq :::"/^"::: ) (Set "(" (Set (Var "k1")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k16_finseq_1 :::"|"::: ) (Set "(" (Set "(" (Set (Var "k2")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))))) "implies" (Bool (Set ($#k2_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_rfinseq :::"/^"::: ) (Set "(" (Set (Var "k2")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k16_finseq_1 :::"|"::: ) (Set "(" (Set "(" (Set (Var "k1")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k2")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ")" ))); definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "k1", "k2" be ($#m1_hidden :::"Nat":::); :: original: :::"mid"::: redefine func :::"mid"::: "(" "f" "," "k1" "," "k2" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; theorem :: FINSEQ_6:113 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k1"))) & (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))) & (Bool (Set (Var "k2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "k2")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "k1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))))) ; theorem :: FINSEQ_6:114 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" )))))) ; theorem :: FINSEQ_6:115 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "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 "f"))))) "holds" (Bool (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "i")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1) ")" )))))) ; theorem :: FINSEQ_6:116 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Num 1) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "k"))))))) ; theorem :: FINSEQ_6:117 (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 "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set "(" (Set (Var "k")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )))))) ; theorem :: FINSEQ_6:118 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k1"))) & (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))) & (Bool (Set (Var "k2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k1")))) & "(" (Bool (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2")))) "implies" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "k2")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k1")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ))) ")" ) ")" ) ")" & "(" (Bool (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k2")))) "implies" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "k1")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k2")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "k1")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) ")" ")" )))) ; theorem :: FINSEQ_6:119 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_6:120 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Num 1) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; theorem :: FINSEQ_6:121 (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 (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")))))) ; theorem :: FINSEQ_6:122 (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 "k1")) "," (Set (Var "k2")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k1"))) & (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))) & (Bool (Set (Var "k2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "k2")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k1")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "k2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "k1")) ")" ) ($#k3_real_1 :::"+"::: ) (Num 1))) "or" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "k2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "k1")))) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k1")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ))) & (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k1")) ")" ))) & (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k1")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ))) & (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k5_real_1 :::"-"::: ) (Num 1) ")" ) ($#k7_real_1 :::"+"::: ) (Set (Var "k1")) ")" ))) ")" )))) ; theorem :: FINSEQ_6:123 (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 "k")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Num 1) "," (Set (Var "k")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))))) ; theorem :: FINSEQ_6:124 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k1"))) & (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))) & (Bool (Set (Var "k2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))))) ; theorem :: FINSEQ_6:125 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "k")) ")" ))))))) ; theorem :: FINSEQ_6:126 (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 "k1")) "," (Set (Var "k2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k2"))) & (Bool (Set (Var "k1")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "k1")) "," (Set (Var "k2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k1")) ")" ) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "k2")) ")" ")" )))))) ;