:: JORDAN8 semantic presentation begin theorem :: JORDAN8:1 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool (Set ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D"))) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))))) ; theorem :: JORDAN8:2 (Bool "for" (Set (Var "m")) "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")) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "m"))) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))))))) ; theorem :: JORDAN8:3 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))) "holds" (Bool "ex" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i1")) "," (Set (Var "j1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" )) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i2")) "," (Set (Var "j2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Set (Var "j1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "i1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ) "or" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "i2")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ) "or" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "j2")) ($#k2_nat_1 :::"+"::: ) (Num 1))) ")" ) ")" ) ")" )))))) ; theorem :: JORDAN8:4 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (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) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_goboard5 :::"standard"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) ")" ))) ; theorem :: JORDAN8:5 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (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) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))) "holds" (Bool "not" (Bool (Set (Var "f")) "is" ($#v3_funct_1 :::"constant"::: ) )))) ; theorem :: JORDAN8:6 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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 "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "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 (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )) & (Bool "(" "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i1")) "," (Set (Var "j1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i2")) "," (Set (Var "j2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" )) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ))) "holds" (Bool (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "i2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i1")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "j2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "j1")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))))) ; theorem :: JORDAN8:7 (Bool "for" (Set (Var "i")) "," (Set (Var "k")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" ) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)))) ; theorem :: JORDAN8:8 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v2_sppol_1 :::"vertical"::: ) ) "iff" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")))) ")" )) ; theorem :: JORDAN8:9 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v1_sppol_1 :::"horizontal"::: ) ) "iff" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")))) ")" )) ; definitionlet "C" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "n" be ($#m1_hidden :::"Nat":::); func :::"Gauge"::: "(" "C" "," "n" ")" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: JORDAN8:def 1 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) "n" ")" ) ($#k1_nat_1 :::"+"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) it)) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) it))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) "C" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) "C" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) "C" ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) "n" ")" ) ")" ) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k5_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) "C" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) "C" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) "C" ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) "n" ")" ) ")" ) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "j")) ($#k5_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"Gauge"::: JORDAN8:def 1 : (Bool "for" (Set (Var "C")) "being" ($#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 "b3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k1_nat_1 :::"+"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "b3")))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k5_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "j")) ($#k5_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )) ")" ) ")" ) ")" )))); registrationlet "C" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ) -> bbbadV3_RELAT_1() ($#v2_goboard1 :::"X_equal-in-line"::: ) ($#v3_goboard1 :::"Y_equal-in-column"::: ) ; end; registrationlet "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":::); cluster (Set ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ) -> ($#v4_goboard1 :::"Y_increasing-in-line"::: ) ($#v5_goboard1 :::"X_increasing-in-column"::: ) ; end; theorem :: JORDAN8:10 (Bool "for" (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 4)))) ; theorem :: JORDAN8:11 (Bool "for" (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "T")))))) ; theorem :: JORDAN8:12 (Bool "for" (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "T")))))) ; theorem :: JORDAN8:13 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 2) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "T")))))) ; theorem :: JORDAN8:14 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "T")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "T")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "T")))))) ; theorem :: JORDAN8:15 (Bool "for" (Set (Var "C")) "being" ($#~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 "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))))) ; theorem :: JORDAN8:16 (Bool "for" (Set (Var "C")) "being" ($#~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 "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))))) ; theorem :: JORDAN8:17 (Bool "for" (Set (Var "C")) "being" ($#~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 "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))))) ; theorem :: JORDAN8:18 (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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "j")) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))))) ;