:: GOBOARD7 semantic presentation begin theorem :: GOBOARD7:1 (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" "not" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Var "s"))) "or" (Bool (Set (Set (Var "r1")) ($#k7_real_1 :::"+"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) "or" (Bool (Set (Set (Var "r2")) ($#k7_real_1 :::"+"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) ")" )) ; theorem :: GOBOARD7:2 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "r")) ($#k9_real_1 :::"-"::: ) (Set (Var "s")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) ")" )) ; theorem :: GOBOARD7:3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p1"))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ; theorem :: GOBOARD7:4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "p1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ; theorem :: GOBOARD7:5 (Bool "for" (Set (Var "p1")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ))) "holds" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: GOBOARD7:6 (Bool "for" (Set (Var "p1")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: GOBOARD7:7 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "p1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p1")) ")" ))) ; theorem :: GOBOARD7:8 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "p1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "p1")) ")" ))) ; theorem :: GOBOARD7:9 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ))))) ; theorem :: GOBOARD7:10 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) )) "holds" (Bool "ex" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )))) ; theorem :: GOBOARD7:11 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) )) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m2_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 "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" )))) ; theorem :: GOBOARD7:12 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; begin theorem :: GOBOARD7:13 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GOBOARD7:14 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GOBOARD7:15 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GOBOARD7:16 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GOBOARD7:17 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GOBOARD7:18 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: GOBOARD7:19 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set ($#k1_int_2 :::"abs"::: ) (Set "(" (Set (Var "j1")) ($#k9_real_1 :::"-"::: ) (Set (Var "j2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))) ; theorem :: GOBOARD7:20 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) & (Bool (Set ($#k1_int_2 :::"abs"::: ) (Set "(" (Set (Var "i1")) ($#k9_real_1 :::"-"::: ) (Set (Var "i2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) ")" ))) ; theorem :: GOBOARD7:21 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) "or" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1))) ")" ) & (Bool "(" (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) "or" (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ) ")" ))) ; theorem :: GOBOARD7:22 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) & (Bool "not" (Bool "(" (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ")" )) & (Bool "not" (Bool "(" (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) ; theorem :: GOBOARD7:23 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" )) & (Bool "not" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" )) ")" )) & (Bool "not" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) ; theorem :: GOBOARD7:24 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" )) & (Bool "not" (Bool "(" (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k1_tarski :::"}"::: ) )) ")" ))) ; theorem :: GOBOARD7:25 (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ))) ; theorem :: GOBOARD7:26 (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ))) ; theorem :: GOBOARD7:27 (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool "for" (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) "or" "not" (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) "or" "not" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) "or" "not" (Bool (Set (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) "or" "not" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ")" )))) ; theorem :: GOBOARD7:28 (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool "for" (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) "or" "not" (Bool (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) "or" "not" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) "or" "not" (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) "or" "not" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j2")) ")" ")" ) ")" )) ")" )))) ; begin theorem :: GOBOARD7:29 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ; theorem :: GOBOARD7:30 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ))) ; theorem :: GOBOARD7:31 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ))) ; theorem :: GOBOARD7:32 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Num 1))) ; theorem :: GOBOARD7:33 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Num 1))) ; theorem :: GOBOARD7:34 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) ; theorem :: GOBOARD7:35 (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")))))) ; theorem :: GOBOARD7:36 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")))))) ; theorem :: GOBOARD7:37 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")))))) ; theorem :: GOBOARD7:38 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "being" ($#m2_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 (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD7:39 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) ")" )))) ; theorem :: GOBOARD7:40 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) ")" )))) ; theorem :: GOBOARD7:41 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:42 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:43 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:44 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ))) ; theorem :: GOBOARD7:45 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:46 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:47 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ))) ; theorem :: GOBOARD7:48 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ))) ; theorem :: GOBOARD7:49 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ))) ; theorem :: GOBOARD7:50 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:51 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: GOBOARD7:52 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ))) ; theorem :: GOBOARD7:53 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" )))) ; theorem :: GOBOARD7:54 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" )))) ; theorem :: GOBOARD7:55 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" ) ")" ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" ) ")" )))) ; theorem :: GOBOARD7:56 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" )))) ; theorem :: GOBOARD7:57 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" )))) ; theorem :: GOBOARD7:58 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool "not" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) ")" ) ")" )))) ; theorem :: GOBOARD7:59 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD7:60 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD7:61 (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD7:62 (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD7:63 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "p1")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "q1")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) ) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "q1")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ))) ; theorem :: GOBOARD7:64 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "p1")) "," (Set (Var "q1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "q1")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) ) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "q1")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ))) ;