:: MATRIX13 semantic presentation begin theorem :: MATRIX13:1 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) ")" ) ")" )))) ; theorem :: MATRIX13:2 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Lower_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) "iff" (Bool (Set (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ) "is" ($#m1_matrix_1 :::"Upper_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) ")" )))) ; theorem :: MATRIX13:3 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set "(" (Set (Var "M")) ($#k5_matrix_1 :::"@"::: ) ")" )))))) ; theorem :: MATRIX13:4 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "perm")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "perm")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n"))))) "holds" (Bool "(" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::">"::: ) (Set (Var "i"))) ")" )) & (Bool "ex" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "perm")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) ")" )) ")" ))) ; theorem :: MATRIX13:5 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "perm")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "perm")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")))) & (Bool "(" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Lower_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) "or" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Upper_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) ")" )) "holds" (Bool (Set (Set "(" ($#k11_matrix_3 :::"Path_product"::: ) (Set (Var "M")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "perm"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX13:6 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "I")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "I")) "," (Set (Var "M")) ")" )))))) ; theorem :: MATRIX13:7 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Upper_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k1_finsop_1 :::"$$"::: ) (Set "(" ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M")) ")" )))))) ; theorem :: MATRIX13:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Lower_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k1_finsop_1 :::"$$"::: ) (Set "(" ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M")) ")" )))))) ; theorem :: MATRIX13:9 (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "Y")) where Y "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) : (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) "}" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "X")) ")" ) ($#k6_newton :::"choose"::: ) (Set (Var "n")))))) ; theorem :: MATRIX13:10 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k2_sgraph1 :::"2Set"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Num 2)))) ; theorem :: MATRIX13:11 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "R")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_matrix_2 :::"even"::: ) ) "iff" (Bool (Set (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Num 2) ")" ) ($#k4_nat_d :::"mod"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: MATRIX13:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" ))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) "holds" (Bool (Set (Set (Var "M")) ($#k4_matrix11 :::"*"::: ) (Set (Var "R"))) "is" ($#m1_matrix_1 :::"Upper_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))))))) ; theorem :: MATRIX13:13 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" ))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) "holds" (Bool (Set (Set (Var "M")) ($#k4_matrix11 :::"*"::: ) (Set (Var "R"))) "is" ($#m1_matrix_1 :::"Lower_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))))))) ; theorem :: MATRIX13:14 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "R")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" ))) & (Bool "(" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "or" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "j"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) ")" )) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k14_matrix_2 :::"-"::: ) "(" (Set "(" (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k1_finsop_1 :::""**""::: ) (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "R")) "," (Set (Var "M")) ")" ")" ) ")" ) "," (Set (Var "R")) ")" )))))) ; theorem :: MATRIX13:15 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Upper_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k4_matrix_6 :::"*"::: ) (Set (Var "M2"))))) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Upper_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) & (Bool (Set ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set "(" ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M2")) ")" ) ")" )) ")" ))))) ; theorem :: MATRIX13:16 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Lower_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k4_matrix_6 :::"*"::: ) (Set (Var "M2"))))) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Lower_Triangular_Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K"))) & (Bool (Set ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set "(" ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k13_matrix_3 :::"diagonal_of_Matrix"::: ) (Set (Var "M2")) ")" ) ")" )) ")" ))))) ; begin definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); let "n", "m" be ($#m1_hidden :::"Nat":::); let "nt" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); let "mt" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); func :::"Segm"::: "(" "M" "," "nt" "," "mt" ")" -> ($#m1_matrix_1 :::"Matrix"::: ) "of" "n" "," "m" "," "D" means :: MATRIX13:def 1 (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")) ")" ) ($#r1_hidden :::"="::: ) (Set "M" ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" "nt" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" "mt" ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ")" ))); end; :: deftheorem defines :::"Segm"::: MATRIX13:def 1 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "b7")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "b7")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" )) "iff" (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "b7"))))) "holds" (Bool (Set (Set (Var "b7")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ")" ))) ")" ))))))); theorem :: MATRIX13:17 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))))) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ))) "iff" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) ")" )))))) ; theorem :: MATRIX13:18 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "not" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) & "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Bool "not" (Set (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "mt")) "," (Set (Var "nt")) ")" ))) ")" ))))))) ; theorem :: MATRIX13:19 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) & "(" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "holds" (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "mt")) "," (Set (Var "nt")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ))))))) ; theorem :: MATRIX13:20 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 1) "," (Set (Var "D")) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_matrlin :::"<*"::: ) (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#k2_matrlin :::"*>"::: ) )))) ; theorem :: MATRIX13:21 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_matrlin :::"<*"::: ) (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) ")" ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#k2_matrlin :::"*>"::: ) ))))))) ; theorem :: MATRIX13:22 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 2) "," (Set (Var "D")) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "(" (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) "," (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 2) ")" ")" ) ")" )))) ; theorem :: MATRIX13:23 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 2))) "holds" (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set "(" (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Num 2) ")" ) ")" ")" ) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ) ")" ")" ) "," (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Num 2) ")" ) ")" ")" ) ")" ))))))) ; theorem :: MATRIX13:24 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A")) "," (Set "(" (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "mt"))))))))) ; theorem :: MATRIX13:25 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "j")) ")" ))))))) ; theorem :: MATRIX13:26 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt")) "," (Set (Var "nt1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "nt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "nt1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX13:27 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt")) "," (Set (Var "nt1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Bool "not" (Set (Var "nt")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "nt1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX13:28 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "A")) "," (Set "(" (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "nt"))))))))) ; theorem :: MATRIX13:29 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "m")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "j")) ")" ))))))) ; theorem :: MATRIX13:30 (Bool "for" (Set (Var "i")) "," (Set (Var "m")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "mt")) "," (Set (Var "mt1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "mt")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "mt1")) "," (Set (Var "mt")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX13:31 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "mt")) "," (Set (Var "mt1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Bool "not" (Set (Var "mt")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "mt1")) "," (Set (Var "mt")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX13:32 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "nt")) "," (Set (Var "nt1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "nt")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "nt1")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt1"))))) "holds" (Bool "ex" (Set (Var "perm")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "st" (Bool (Set (Var "nt1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "nt")) ($#k1_partfun1 :::"*"::: ) (Set (Var "perm"))))))) ; theorem :: MATRIX13:33 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt1")) "," (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "nt1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "nt")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt1")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) ($#k4_matrix11 :::"*"::: ) (Set (Var "f")))))))))) ; theorem :: MATRIX13:34 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt1")) "," (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "mt1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "mt")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt1")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")))))))))) ; theorem :: MATRIX13:35 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt1")) "," (Set (Var "nt2")) "," (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "perm")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "nt1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "nt2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "perm"))))) "holds" (Bool "(" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt1")) "," (Set (Var "nt")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k14_matrix_2 :::"-"::: ) "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt2")) "," (Set (Var "nt")) ")" ")" ) ")" ) "," (Set (Var "perm")) ")" )) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "nt1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k14_matrix_2 :::"-"::: ) "(" (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "nt2")) ")" ")" ) ")" ) "," (Set (Var "perm")) ")" )) ")" )))))) ; theorem :: MATRIX13:36 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "nt")) "," (Set (Var "nt1")) "," (Set (Var "nt9")) "," (Set (Var "nt19")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt9")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt1"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt19")))) & (Bool (Bool "not" (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "nt1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt9")) "," (Set (Var "nt19")) ")" ")" ))))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "nt1")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt9")) "," (Set (Var "nt19")) ")" ")" ) ")" ))))))) ; theorem :: MATRIX13:37 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "D")) (Bool "for" (Set (Var "F")) "," (Set (Var "Fmt")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A9")))) & (Bool (Set (Var "Fmt")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_partfun1 :::"*"::: ) (Set (Var "mt")))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A9"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "nt")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "j")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k3_matrix11 :::"RLine"::: ) "(" (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A9")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "Fmt")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "A9")) "," (Set (Var "j")) "," (Set (Var "F")) ")" ")" ) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ))))))))) ; theorem :: MATRIX13:38 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "D")) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt"))))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A9"))))) "holds" (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A9")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "A9")) "," (Set (Var "i")) "," (Set (Var "F")) ")" ")" ) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ))))))))) ; theorem :: MATRIX13:39 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A9"))))) "holds" (Bool "ex" (Set (Var "nt1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "i")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "j")) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "A9")) "," (Set (Var "i")) "," (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A9")) "," (Set (Var "j")) ")" ")" ) ")" ")" ) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A9")) "," (Set (Var "nt1")) "," (Set (Var "mt")) ")" )) ")" ))))))) ; theorem :: MATRIX13:40 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "D")) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A9")) ")" ))))) "holds" (Bool (Set ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "A9")) "," (Set (Var "i")) "," (Set (Var "F")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "A9"))))))) ; definitionlet "n", "m" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "m")) "," (Set (Const "K")); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "K")); :: original: :::"*"::: redefine func "a" :::"*"::: "M" -> ($#m1_matrix_1 :::"Matrix":::) "of" "n" "," "m" "," "K"; end; theorem :: MATRIX13:41 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "a")) ($#k2_matrix13 :::"*"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" )))))))) ; theorem :: MATRIX13:42 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "nt")) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ))) & (Bool (Set (Var "mt")) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "A")))))))) ; registration cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); end; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> ($#v1_setfam_1 :::"without_zero"::: ) ; end; registrationlet "X" be ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_hidden :::"set"::: ) ; let "Y" be ($#m1_hidden :::"set"::: ) ; cluster (Set "X" ($#k4_xboole_0 :::"\"::: ) "Y") -> ($#v1_setfam_1 :::"without_zero"::: ) ; end; definitionlet "i" be ($#m1_hidden :::"Nat":::); :: original: :::"{"::: redefine func :::"{":::"i":::"}"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "j" be ($#m1_hidden :::"Nat":::); :: original: :::"{"::: redefine func :::"{":::"i" "," "j":::"}"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: MATRIX13:43 (Bool "for" (Set (Var "N")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) ; definitionlet "N" be ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); :: original: :::"Sgm"::: redefine func :::"Sgm"::: "N" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "(" ($#k5_card_1 :::"card"::: ) "N" ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); let "P", "Q" be ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"Segm"::: "(" "A" "," "P" "," "Q" ")" -> ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set ($#k5_card_1 :::"card"::: ) "P") "," (Set ($#k5_card_1 :::"card"::: ) "Q") "," "D" equals :: MATRIX13:def 2 (Set ($#k1_matrix13 :::"Segm"::: ) "(" "A" "," (Set "(" ($#k5_matrix13 :::"Sgm"::: ) "P" ")" ) "," (Set "(" ($#k5_matrix13 :::"Sgm"::: ) "Q" ")" ) ")" ); end; :: deftheorem defines :::"Segm"::: MATRIX13:def 2 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) "," (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ) ")" ))))); theorem :: MATRIX13:44 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i0")) "," (Set (Var "j0")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i0")) ($#k3_matrix13 :::"}"::: ) ) "," (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "j0")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_matrlin :::"<*"::: ) (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i0")) "," (Set (Var "j0")) ")" ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#k2_matrlin :::"*>"::: ) ))))) ; theorem :: MATRIX13:45 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i0")) "," (Set (Var "j0")) "," (Set (Var "n0")) "," (Set (Var "m0")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j0"))) & (Bool (Set (Var "n0")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m0")))) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "i0")) "," (Set (Var "j0")) ($#k4_matrix13 :::"}"::: ) ) "," (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "n0")) "," (Set (Var "m0")) ($#k4_matrix13 :::"}"::: ) ) ")" ) ($#r1_hidden :::"="::: ) (Set "(" (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i0")) "," (Set (Var "n0")) ")" ")" ) "," (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i0")) "," (Set (Var "m0")) ")" ")" ) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j0")) "," (Set (Var "n0")) ")" ")" ) "," (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j0")) "," (Set (Var "m0")) ")" ")" ) ")" ))))) ; theorem :: MATRIX13:46 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; theorem :: MATRIX13:47 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "P")) ")" ))) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A")) "," (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ))))))) ; theorem :: MATRIX13:48 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "P")) ")" )))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A")) "," (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ")" )))))) ; theorem :: MATRIX13:49 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "Q")) "," (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Q")) ")" ))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "A")) "," (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ))))))) ; theorem :: MATRIX13:50 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "Q")) ")" )))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) ")" ) "," (Set (Var "Q")) ")" ")" ) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "A")) "," (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ")" )))))) ; theorem :: MATRIX13:51 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_matrlin :::"Del"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) ")" ))))) ; theorem :: MATRIX13:52 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) ")" ) "," (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_matrix_2 :::"DelCol"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ))))) ; theorem :: MATRIX13:53 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "X"))) "is" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) )))) ; theorem :: MATRIX13:54 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "P")))) "holds" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "X")) ")" ) ")" ))))) ; theorem :: MATRIX13:55 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "X")) ")" ) "," (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "Y")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ))))))) ; theorem :: MATRIX13:56 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "P1")) "," (Set (Var "Q")) "," (Set (Var "Q1")) "," (Set (Var "P2")) "," (Set (Var "Q2")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set (Var "P1"))) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set (Var "Q1"))) & (Bool (Set (Var "P2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P1")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "P")))) & (Bool (Set (Var "Q2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q1")) ")" ) ($#k8_relat_1 :::"""::: ) (Set (Var "Q"))))) "holds" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P2")) ")" ) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q2")) ")" ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" ))) & (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" ) "," (Set (Var "P2")) "," (Set (Var "Q2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" )) ")" )))) ; theorem :: MATRIX13:57 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "," (Set (Var "P1")) "," (Set (Var "Q1")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "not" (Bool "(" "(" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" ))) & (Bool "(" "for" (Set (Var "P2")) "," (Set (Var "Q2")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Set (Var "P2")) ($#r1_tarski :::"c="::: ) (Set (Var "P1"))) "or" "not" (Bool (Set (Var "Q2")) ($#r1_tarski :::"c="::: ) (Set (Var "Q1"))) "or" "not" (Bool (Set (Var "P2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P1")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "P")))) "or" "not" (Bool (Set (Var "Q2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q1")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "Q")))) "or" "not" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P")))) "or" "not" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) "or" "not" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P2")) "," (Set (Var "Q2")) ")" )) ")" ) ")" ) ")" ))))) ; theorem :: MATRIX13:58 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) "," (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "j")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_2 :::"Deleting"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))))) ; theorem :: MATRIX13:59 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "D")) (Bool "for" (Set (Var "Q")) "," (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "," (Set (Var "FQ")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A9")))) & (Bool (Set (Var "FQ")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A9"))))) "holds" (Bool (Set ($#k3_matrix11 :::"RLine"::: ) "(" (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A9")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "FQ")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "A9")) "," (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set (Var "F")) ")" ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ))))))) ; theorem :: MATRIX13:60 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A9")) "being" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "D")) (Bool "for" (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A9"))))) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A9")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "A9")) "," (Set (Var "i")) "," (Set (Var "F")) ")" ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ))))))))) ; theorem :: MATRIX13:61 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "not" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) & "(" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Bool "not" (Set (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "Q")) "," (Set (Var "P")) ")" ))) ")" ))))) ; theorem :: MATRIX13:62 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) & "(" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "Q")) "," (Set (Var "P")) ")" ")" ) ($#k4_matrix_1 :::"@"::: ) ))))) ; theorem :: MATRIX13:63 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set (Var "a")) ($#k2_matrix13 :::"*"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" )))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "D")); let "P", "Q" be ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); assume (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Const "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Const "Q")))) ; func :::"EqSegm"::: "(" "A" "," "P" "," "Q" ")" -> ($#m1_matrix_1 :::"Matrix":::) "of" (Set ($#k5_card_1 :::"card"::: ) "P") "," "D" equals :: MATRIX13:def 3 (Set ($#k6_matrix13 :::"Segm"::: ) "(" "A" "," "P" "," "Q" ")" ); end; :: deftheorem defines :::"EqSegm"::: MATRIX13:def 3 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool (Set ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "A")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ))))); theorem :: MATRIX13:64 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "P")) ")" ))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "P")) ")" ))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool "(" (Bool (Set ($#k2_laplace :::"Delete"::: ) "(" (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set "(" (Set (Var "P")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_matrix13 :::"}"::: ) ) ")" ) "," (Set "(" (Set (Var "Q")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k3_matrix13 :::"}"::: ) ) ")" ) ")" )) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "P")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "P")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_matrix13 :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set (Var "Q")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set "(" (Set "(" ($#k5_matrix13 :::"Sgm"::: ) (Set (Var "Q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k3_matrix13 :::"}"::: ) ) ")" ))) ")" ))))) ; theorem :: MATRIX13:65 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "P1")) "," (Set (Var "Q1")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q1")))) & (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set (Var "P1"))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "ex" (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set (Var "Q1"))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ))))) ; theorem :: MATRIX13:66 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P1")) "," (Set (Var "Q")) "," (Set (Var "Q1")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q1")))) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set (Var "Q1"))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "ex" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set (Var "P1"))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ))))) ; theorem :: MATRIX13:67 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) "iff" (Bool "(" (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ))) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ))) ")" ) ")" )))) ; theorem :: MATRIX13:68 (Bool "for" (Set (Var "m9")) "," (Set (Var "n9")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M9")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "j0")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n9")))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "P"))) & (Bool (Bool "not" (Set (Var "j0")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M9"))))) "holds" (Bool "(" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set "(" (Set "(" (Set (Var "P")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "j0")) ($#k3_matrix13 :::"}"::: ) ) ")" ))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set "(" (Set (Var "P")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "j0")) ($#k3_matrix13 :::"}"::: ) ) ")" ) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M9")))) & (Bool "(" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) "," (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M9")) "," (Set (Var "j0")) ")" ")" ) ")" ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M9")) "," (Set "(" (Set "(" (Set (Var "P")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "j0")) ($#k3_matrix13 :::"}"::: ) ) ")" ) "," (Set (Var "Q")) ")" ")" ))) "or" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) "," (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M9")) "," (Set (Var "j0")) ")" ")" ) ")" ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M9")) "," (Set "(" (Set "(" (Set (Var "P")) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "j0")) ($#k3_matrix13 :::"}"::: ) ) ")" ) "," (Set (Var "Q")) ")" ")" ) ")" ))) ")" ) ")" ))))))) ; theorem :: MATRIX13:69 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))) "iff" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "Q")) "," (Set (Var "P")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ))) ")" )))) ; theorem :: MATRIX13:70 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" ) "," (Set (Var "Q")) "," (Set (Var "P")) ")" ")" )))))) ; theorem :: MATRIX13:71 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" (Set (Var "a")) ($#k2_matrix13 :::"*"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set (Var "a")) "," (Set (Var "n")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M")) ")" ))))))) ; theorem :: MATRIX13:72 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_laplace :::"."::: ) "(" (Set (Var "a")) "," (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "P")) ")" ) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" ) ")" ))))))) ; definitionlet "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "K")); func :::"the_rank_of"::: "M" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: MATRIX13:def 4 (Bool "(" (Bool "ex" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M")) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) it) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" "M" "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "K")) ")" )) & (Bool "(" "for" (Set (Var "P1")) "," (Set (Var "Q1")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P1")) "," (Set (Var "Q1")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) "M")) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q1")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" "M" "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "K"))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_xxreal_0 :::"<="::: ) it) ")" ) ")" ); end; :: deftheorem defines :::"the_rank_of"::: MATRIX13:def 4 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M")))) "iff" (Bool "(" (Bool "ex" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set (Var "b3"))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) & (Bool "(" "for" (Set (Var "P1")) "," (Set (Var "Q1")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P1")) "," (Set (Var "Q1")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q1")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b3"))) ")" ) ")" ) ")" )))); theorem :: MATRIX13:73 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))))) "holds" (Bool "(" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) ")" )))) ; theorem :: MATRIX13:74 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) & (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")))) ")" ))) ; theorem :: MATRIX13:75 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt1")) "," (Set (Var "nt2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt1")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt2")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt1")) "," (Set (Var "nt2")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "ex" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "P1")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt1")))) & (Bool (Set (Var "P2")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt2")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P2")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P1")) "," (Set (Var "P2")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )))))) ; theorem :: MATRIX13:76 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "RANK")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "RANK"))) "iff" (Bool "(" (Bool "ex" (Set (Var "rt1")) "," (Set (Var "rt2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "RANK")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool "(" (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "rt1")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "rt2")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "rt1")) "," (Set (Var "rt2")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "nt1")) "," (Set (Var "nt2")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt1")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt2")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt1")) "," (Set (Var "nt2")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "RANK")))) ")" ) ")" ) ")" )))) ; theorem :: MATRIX13:77 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))))))) ; theorem :: MATRIX13:78 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" )))))))) ; theorem :: MATRIX13:79 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )))))) ; theorem :: MATRIX13:80 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "," (Set (Var "P1")) "," (Set (Var "Q")) "," (Set (Var "Q1")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set (Var "P1"))) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set (Var "Q1")))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P1")) "," (Set (Var "Q1")) ")" ")" )))))) ; theorem :: MATRIX13:81 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set (Var "h")))) ")" ))) ; theorem :: MATRIX13:82 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "nt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "mt")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "m")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "nt")) ")" ) "," (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "mt")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k1_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "nt")) "," (Set (Var "mt")) ")" ")" )))))))) ; theorem :: MATRIX13:83 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) "iff" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )))) ; theorem :: MATRIX13:84 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" (Set (Var "M")) ($#k4_matrix_1 :::"@"::: ) ")" ))))) ; theorem :: MATRIX13:85 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "K")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" (Set (Var "M")) ($#k4_matrix11 :::"*"::: ) (Set (Var "F")) ")" ))))))) ; theorem :: MATRIX13:86 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "M")) ")" )))))) ; theorem :: MATRIX13:87 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "p")) "," (Set (Var "pf")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "pf")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "pf")))))))) ; theorem :: MATRIX13:88 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "p")) "," (Set (Var "pf")) "," (Set (Var "q")) "," (Set (Var "qf")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "pf")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "qf")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k3_relat_1 :::"*"::: ) (Set (Var "f")))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k3_fvsum_1 :::"+"::: ) (Set (Var "q")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "pf")) ($#k3_fvsum_1 :::"+"::: ) (Set (Var "qf"))))))) ; theorem :: MATRIX13:89 (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M9")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M9"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) "," (Set "(" (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) ")" ")" ) ")" ) ")" ")" ))))))) ; theorem :: MATRIX13:90 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ($#k5_finseq_2 :::"|->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" )))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k8_matrix_2 :::"DelLine"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))))))) ; theorem :: MATRIX13:91 (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M9")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "K")) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M9"))))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k8_matrix_2 :::"DelLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" ) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "p")) ")" ) ")" ")" ))))))) ; theorem :: MATRIX13:92 (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M9")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M9")) ")" ))) & "(" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))) "implies" (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k1_group_1 :::"1_"::: ) (Set (Var "K")) ")" ))) ")" ) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M9"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) ")" ")" ) ($#k4_fvsum_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M9")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ")" ")" ))))))) ; theorem :: MATRIX13:93 (Bool "for" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M9")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n9")) "," (Set (Var "m9")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M9")) ")" ))) & (Bool (Set (Var "j")) ($#r1_hidden :::"<>"::: ) (Set (Var "i")))) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k8_matrix_2 :::"DelLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M9")) "," (Set (Var "i")) "," (Set "(" (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M9")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ")" ))))))) ; theorem :: MATRIX13:94 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool "ex" (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) ")" ))) ; theorem :: MATRIX13:95 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" )) ")" ))) ; theorem :: MATRIX13:96 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool "(" (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#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 "M")))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) & (Bool "(" "for" (Set (Var "i0")) "," (Set (Var "j0")) "," (Set (Var "n0")) "," (Set (Var "m0")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i0")) ($#r1_hidden :::"<>"::: ) (Set (Var "j0"))) & (Bool (Set (Var "n0")) ($#r1_hidden :::"<>"::: ) (Set (Var "m0"))) & (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "i0")) "," (Set (Var "j0")) ($#k4_matrix13 :::"}"::: ) ) "," (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "n0")) "," (Set (Var "m0")) ($#k4_matrix13 :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "i0")) "," (Set (Var "j0")) ($#k4_matrix13 :::"}"::: ) ) "," (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "n0")) "," (Set (Var "m0")) ($#k4_matrix13 :::"}"::: ) ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ) ")" ))) ; theorem :: MATRIX13:97 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool "(" (Bool "ex" (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_matrix13 :::"}"::: ) ) "," (Set ($#k4_matrix13 :::"{"::: ) (Set (Var "n")) "," (Set (Var "m")) ($#k4_matrix13 :::"}"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "m")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "n")) ")" ")" ))) ")" ) ")" ) ")" ))) ; theorem :: MATRIX13:98 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ))) & (Bool "ex" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" )))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )))) ")" ) ")" )) ")" ))) ; registrationlet "K" be ($#l6_algstr_0 :::"Field":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K") ($#k3_finseq_2 :::"*"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_funcop_1 :::"Function-yielding"::: ) bbbadV2_FUNCOP_1() ($#v1_matrix_1 :::"tabular"::: ) ($#v2_matrix_1 :::"diagonal"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K") ($#k3_finseq_2 :::"*"::: ) ); end; theorem :: MATRIX13:99 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#v2_matrix_1 :::"diagonal"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "NonZero1")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "NonZero1")) ($#r1_hidden :::"="::: ) "{" (Set (Var "i")) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) "}" )) "holds" (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "Q")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "Q")))) & (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" ($#k7_matrix13 :::"EqSegm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "Q")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "(" (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set (Var "NonZero1"))) & (Bool (Set (Var "Q")) ($#r1_tarski :::"c="::: ) (Set (Var "NonZero1"))) ")" ))))) ; theorem :: MATRIX13:100 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#v2_matrix_1 :::"diagonal"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "P")) ($#k2_zfmisc_1 :::":]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set (Var "P")) ")" ) "is" bbbadV2_MATRIX_1((Set (Var "b1"))))))) ; theorem :: MATRIX13:101 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#v2_matrix_1 :::"diagonal"::: ) ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "NonZero1")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "NonZero1")) ($#r1_hidden :::"="::: ) "{" (Set (Var "i")) where i "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "i")) "," (Set (Var "i")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "M")))) & (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) "}" )) "holds" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "NonZero1"))))))) ; theorem :: MATRIX13:102 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" ) (Bool "for" (Set (Var "t1")) "," (Set (Var "t2")) "," (Set (Var "t")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K")))) "holds" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K"))))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" ))) & "(" (Bool (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set (Var "v1"))) & (Bool (Set (Var "t2")) ($#r1_hidden :::"="::: ) (Set (Var "v2")))) "implies" (Bool (Set (Set (Var "t1")) ($#k3_fvsum_1 :::"+"::: ) (Set (Var "t2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v1")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v2")))) ")" & "(" (Bool (Bool (Set (Var "t")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "implies" (Bool (Set (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")))) ")" ")" )))))) ; registrationlet "K" be ($#l6_algstr_0 :::"Field":::); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set "n" ($#k6_prvect_1 :::"-VectSp_over"::: ) "K") -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ; end; registrationlet "K" be ($#l6_algstr_0 :::"Field":::); let "n" be ($#m1_hidden :::"Nat":::); cluster -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" "n" ($#k6_prvect_1 :::"-VectSp_over"::: ) "K" ")" )); end; notationlet "K" be ($#l6_algstr_0 :::"Field":::); let "m", "n" be ($#m1_hidden :::"Nat":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "m")) "," (Set (Const "n")) "," (Set (Const "K")); synonym :::"lines"::: "M" for :::"rng"::: "K"; synonym :::"without_repeated_line"::: "M" for :::"one-to-one"::: ; end; definitionlet "K" be ($#l6_algstr_0 :::"Field":::); let "m", "n" be ($#m1_hidden :::"Nat":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "m")) "," (Set (Const "n")) "," (Set (Const "K")); :: original: :::"lines"::: redefine func :::"lines"::: "M" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "n" ($#k6_prvect_1 :::"-VectSp_over"::: ) "K" ")" ); end; theorem :: MATRIX13:103 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M")))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" )) ")" )) ")" ))))) ; theorem :: MATRIX13:104 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "V")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" ) (Bool "ex" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set ($#k5_card_1 :::"card"::: ) (Set (Var "V"))) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) & (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "V"))) ")" ))))) ; definitionlet "K" be ($#l6_algstr_0 :::"Field":::); let "n" be ($#m1_hidden :::"Nat":::); let "F" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" (Set (Const "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Const "K")) ")" ); func :::"FinS2MX"::: "F" -> ($#m1_matrix_1 :::"Matrix":::) "of" (Set ($#k3_finseq_1 :::"len"::: ) "F") "," "n" "," "K" equals :: MATRIX13:def 5 "F"; end; :: deftheorem defines :::"FinS2MX"::: MATRIX13:def 5 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" ) "holds" (Bool (Set ($#k10_matrix13 :::"FinS2MX"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Var "F")))))); definitionlet "K" be ($#l6_algstr_0 :::"Field":::); let "m", "n" be ($#m1_hidden :::"Nat":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "m")) "," (Set (Const "n")) "," (Set (Const "K")); func :::"MX2FinS"::: "M" -> ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" "n" ($#k6_prvect_1 :::"-VectSp_over"::: ) "K" ")" ) equals :: MATRIX13:def 6 "M"; end; :: deftheorem defines :::"MX2FinS"::: MATRIX13:def 6 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k11_matrix13 :::"MX2FinS"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "M")))))); theorem :: MATRIX13:105 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "m")))) "holds" (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) )))) ; theorem :: MATRIX13:106 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "L")) "being" ($#m1_vectsp_6 :::"Linear_Combination"::: ) "of" (Set (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K"))) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ))) & (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "M")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" ($#k10_matrix13 :::"FinS2MX"::: ) (Set "(" (Set (Var "L")) ($#k3_vectsp_6 :::"(#)"::: ) (Set "(" ($#k11_matrix13 :::"MX2FinS"::: ) (Set (Var "M")) ")" ) ")" ) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )))))))) ; theorem :: MATRIX13:107 (Bool "for" (Set (Var "m")) "," (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "L")) "being" ($#m1_vectsp_6 :::"Linear_Combination"::: ) "of" (Set (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K"))) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) & (Bool (Set ($#k1_vectsp_6 :::"Carrier"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M")))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k4_vectsp_6 :::"Sum"::: ) (Set (Var "L")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" ($#k10_matrix13 :::"FinS2MX"::: ) (Set "(" (Set (Var "L")) ($#k3_vectsp_6 :::"(#)"::: ) (Set "(" ($#k11_matrix13 :::"MX2FinS"::: ) (Set (Var "M")) ")" ) ")" ) ")" ) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: MATRIX13:108 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "," (Set (Var "M1")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M1")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )))) ")" )) "holds" (Bool "ex" (Set (Var "L")) "being" ($#m2_vectsp_6 :::"Linear_Combination"::: ) "of" (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "st" (Bool (Set (Set (Var "L")) ($#k3_vectsp_6 :::"(#)"::: ) (Set "(" ($#k11_matrix13 :::"MX2FinS"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "M1"))))))) ; theorem :: MATRIX13:109 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) )) "holds" (Bool "(" (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "n")) ($#k5_finseq_2 :::"|->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "M1")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M1")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )))) ")" ) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M1")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "m")) "," (Set (Var "n")) ")" )) ")" ) ")" ) "iff" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) ")" )))) ; theorem :: MATRIX13:110 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) )))) ; theorem :: MATRIX13:111 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Diagonal":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#m1_vectsp_7 :::"Basis"::: ) "of" (Set (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K"))))))) ; registrationlet "K" be ($#l6_algstr_0 :::"Field":::); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set "n" ($#k6_prvect_1 :::"-VectSp_over"::: ) "K") -> ($#v1_matrlin :::"finite-dimensional"::: ) ; end; theorem :: MATRIX13:112 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) "holds" (Bool (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "n"))))) ; theorem :: MATRIX13:113 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" )) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) "iff" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) ")" ) "," (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ")" ) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) ")" )))))) ; theorem :: MATRIX13:114 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) & (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )) "holds" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) ")" ) "," (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k3_matrix13 :::"{"::: ) (Set (Var "i")) ($#k3_matrix13 :::"}"::: ) ) ")" ) ")" ")" )) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ))))) ; theorem :: MATRIX13:115 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "U")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "U")) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) )) "holds" (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "U"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "U"))) & (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set (Var "v")))) "holds" (Bool (Set (Set "(" (Set (Var "U")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "u")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) )))))) ; theorem :: MATRIX13:116 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_vectsp_7 :::"Lin"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "u")) "," (Set (Var "v")) ($#k7_domain_1 :::"}"::: ) ))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "u")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "v")) ")" )))) ")" ))))) ; theorem :: MATRIX13:117 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) & (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) )) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ))) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool "(" (Bool (Set ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k4_fvsum_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ")" ) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) & (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set "(" ($#k3_matrix11 :::"RLine"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) "," (Set "(" (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k4_fvsum_1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ")" ")" )) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) ")" )))))) ; theorem :: MATRIX13:118 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M")))))))) ; theorem :: MATRIX13:119 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) )) "holds" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ")" )) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ))))) ; theorem :: MATRIX13:120 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) )) "holds" (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ))))) ; theorem :: MATRIX13:121 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) & (Bool (Set (Var "M")) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) ")" ) "iff" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) ")" )))) ; theorem :: MATRIX13:122 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "U")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" ) "st" (Bool (Bool (Set (Var "U")) ($#r1_tarski :::"c="::: ) (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))))) "holds" (Bool "ex" (Set (Var "P")) "being" ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"without_zero"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "P")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool (Set ($#k9_matrix13 :::"lines"::: ) (Set "(" ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "U"))) & (Bool (Set ($#k6_matrix13 :::"Segm"::: ) "(" (Set (Var "M")) "," (Set (Var "P")) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ")" ) "is" ($#v2_funct_1 :::"without_repeated_line"::: ) ) ")" )))))) ; theorem :: MATRIX13:123 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "RANK")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k8_matrix13 :::"the_rank_of"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Var "RANK"))) "iff" (Bool "(" (Bool "ex" (Set (Var "U")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" ) "st" (Bool "(" (Bool (Set (Var "U")) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) & (Bool (Set (Var "U")) ($#r1_tarski :::"c="::: ) (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M")))) & (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "U"))) ($#r1_hidden :::"="::: ) (Set (Var "RANK"))) ")" )) & (Bool "(" "for" (Set (Var "W")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "n")) ($#k6_prvect_1 :::"-VectSp_over"::: ) (Set (Var "K")) ")" ) "st" (Bool (Bool (Set (Var "W")) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) ) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set ($#k9_matrix13 :::"lines"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set (Var "W"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "RANK"))) ")" ) ")" ) ")" ))))) ;