:: JORDAN9 semantic presentation begin theorem :: JORDAN9:1 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "B")) "," (Set (Var "C1")) "," (Set (Var "C2")) "," (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "C1")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Var "D"))) & (Bool (Set (Var "C2")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Var "D"))) & (Bool (Set (Var "B")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C1"))) & (Bool (Set (Var "B")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C2"))) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "C1")) ($#r1_hidden :::"="::: ) (Set (Var "C2"))))) ; theorem :: JORDAN9:2 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: JORDAN9:3 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")))) ")" ))))) ; theorem :: JORDAN9:4 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")))) ")" ))))) ; begin theorem :: JORDAN9:5 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) "iff" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f"))) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) ")" )))) ; theorem :: JORDAN9:6 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "G"))))))) ; theorem :: JORDAN9:7 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" )) ")" ))))) ; theorem :: JORDAN9:8 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" )) ")" )))) ; theorem :: JORDAN9:9 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )))) ; theorem :: JORDAN9:10 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" )) "is" ($#v1_convex1 :::"convex"::: ) ) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" )) "is" ($#v1_convex1 :::"convex"::: ) ) ")" )))) ; theorem :: JORDAN9:11 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" )) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" )) ")" )))) ; theorem :: JORDAN9:12 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) )) "holds" (Bool "ex" (Set (Var "j")) "being" ($#m1_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 (Var "G")))) & (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 (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ))))) ; theorem :: JORDAN9:13 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) )) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool "(" "for" (Set (Var "p")) "being" ($#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"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" ))))) ; theorem :: JORDAN9:14 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: JORDAN9:15 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) ")" )))) ; theorem :: JORDAN9:16 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_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 "(" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ))) ; theorem :: JORDAN9:17 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#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 ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) "iff" (Bool "(" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )))) ; theorem :: JORDAN9:18 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_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 ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) ) where r, s "is" ($#m1_subset_1 :::"Real":::) : (Bool "(" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) "}" ))) ; theorem :: JORDAN9:19 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "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")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "G")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ))))) ; theorem :: JORDAN9:20 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#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 ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool "(" (Bool (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ))) ; theorem :: JORDAN9:21 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "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")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "G")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Var "p")) ($#r1_sppol_1 :::"is_extremal_in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))))) ; theorem :: JORDAN9:22 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (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"))))) "holds" (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "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")))) & (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" ))))) ; theorem :: JORDAN9:23 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k16_matrix_1 :::"Values"::: ) (Set (Var "G")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" )) & (Bool (Bool "not" (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))))) ; theorem :: JORDAN9:24 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")) ")" ) "," (Set (Var "k")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN9:25 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN9:26 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (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"))))) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) "or" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" )) "holds" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ))))) ; theorem :: JORDAN9:27 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (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 (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))) "holds" (Bool "for" (Set (Var "k")) "being" ($#m1_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 ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" )))) ; begin theorem :: JORDAN9:28 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" )))) ; theorem :: JORDAN9:29 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "i1")) "," (Set (Var "i2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Set (Var "i1")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i1")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "i2")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i2")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))))) ; theorem :: JORDAN9:30 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_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 ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))) & (Bool (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C"))) ")" ) ")" ) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" ))) "holds" (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)))))) ; definitionlet "C" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "n" be ($#m1_hidden :::"Nat":::); assume (Bool (Set (Const "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) ; func :::"Cage"::: "(" "C" "," "n" ")" -> ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) means :: JORDAN9:def 1 (Bool "(" (Bool it ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" )) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ))) & (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ")" )) & (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) "C") ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) "C") ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) it))) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k5_gobrd13 :::"front_left_cell"::: ) "(" it "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) "C") & (Bool (Set ($#k4_gobrd13 :::"front_right_cell"::: ) "(" it "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) "C")) "implies" (Bool it ($#r1_gobrd13 :::"turns_right"::: ) (Set (Var "k")) "," (Set ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" )) ")" & "(" (Bool (Bool (Set ($#k5_gobrd13 :::"front_left_cell"::: ) "(" it "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) "C") & (Bool (Set ($#k4_gobrd13 :::"front_right_cell"::: ) "(" it "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) "C")) "implies" (Bool it ($#r3_gobrd13 :::"goes_straight"::: ) (Set (Var "k")) "," (Set ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" )) ")" & "(" (Bool (Bool (Set ($#k5_gobrd13 :::"front_left_cell"::: ) "(" it "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) "C")) "implies" (Bool it ($#r2_gobrd13 :::"turns_left"::: ) (Set (Var "k")) "," (Set ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" )) ")" ")" ) ")" ) ")" ); end; :: deftheorem defines :::"Cage"::: JORDAN9:def 1 : (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) "iff" (Bool "(" (Bool (Set (Var "b3")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b3"))))) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k5_gobrd13 :::"front_left_cell"::: ) "(" (Set (Var "b3")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_gobrd13 :::"front_right_cell"::: ) "(" (Set (Var "b3")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C")))) "implies" (Bool (Set (Var "b3")) ($#r1_gobrd13 :::"turns_right"::: ) (Set (Var "k")) "," (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) ")" & "(" (Bool (Bool (Set ($#k5_gobrd13 :::"front_left_cell"::: ) "(" (Set (Var "b3")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_gobrd13 :::"front_right_cell"::: ) "(" (Set (Var "b3")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C")))) "implies" (Bool (Set (Var "b3")) ($#r3_gobrd13 :::"goes_straight"::: ) (Set (Var "k")) "," (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) ")" & "(" (Bool (Bool (Set ($#k5_gobrd13 :::"front_left_cell"::: ) "(" (Set (Var "b3")) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C")))) "implies" (Bool (Set (Var "b3")) ($#r2_gobrd13 :::"turns_left"::: ) (Set (Var "k")) "," (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) ")" ")" ) ")" ) ")" ) ")" )))); theorem :: JORDAN9:31 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool "(" (Bool (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))) & (Bool (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C"))) ")" ))) ; theorem :: JORDAN9:32 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) )) "holds" (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1))))) ;