:: MATRIX_4 semantic presentation begin definitionlet "K" be ($#l6_algstr_0 :::"Field":::); let "M1", "M2" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Const "K")); func "M1" :::"-"::: "M2" -> ($#m2_finseq_1 :::"Matrix":::) "of" "K" equals :: MATRIX_4:def 1 (Set "M1" ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) "M2" ")" )); end; :: deftheorem defines :::"-"::: MATRIX_4:def 1 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ))))); theorem :: MATRIX_4:1 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "M"))))) ; theorem :: MATRIX_4:2 (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 (Set (Var "M")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (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 :: MATRIX_4:3 (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 (Set (Var "M")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" )))) ; theorem :: MATRIX_4:4 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:5 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:6 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M2")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1")) ")" ) ")" )))) ; theorem :: MATRIX_4:7 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1")) ")" ) ")" ))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:8 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1")) ")" ) ")" ))) "holds" (Bool (Set (Var "M2")) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:9 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) "holds" (Bool (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) "," (Set (Var "m")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) "," (Set (Var "m")) ")" )))) ; theorem :: MATRIX_4:10 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set (Var "M2")))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1")) ")" ) ")" )))) ; theorem :: MATRIX_4:11 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:12 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:13 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:14 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:15 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:16 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:17 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:18 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:19 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "," (Set (Var "M4")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M4")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M4")))) & (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M4"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M4")))))) ; theorem :: MATRIX_4:20 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:21 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:22 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:23 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:24 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:25 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:26 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" ))))) ; theorem :: MATRIX_4:27 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M3")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:28 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "," (Set (Var "M4")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M4")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M4")))) & (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M4"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M4")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:29 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "," (Set (Var "M4")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M4")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M4")))) & (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M4")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M4")))))) ; theorem :: MATRIX_4:30 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "," (Set (Var "M4")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M4")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M4")))) & (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M4"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M4"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:31 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M2")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")))))) ; theorem :: MATRIX_4:32 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M3")))))) ; theorem :: MATRIX_4:33 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:34 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" ))))) ; theorem :: MATRIX_4:35 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:36 (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 ($#k2_matrix_3 :::"-"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ))) "holds" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" )))) ; theorem :: MATRIX_4:37 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1")) ")" ) ")" ))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:38 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:39 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "M2")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ")" ))))) ; theorem :: MATRIX_4:40 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M1")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:41 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:42 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ")" ))))) ; theorem :: MATRIX_4:43 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:44 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:45 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:46 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")))))) ; theorem :: MATRIX_4:47 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:48 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M3")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")))))) ; theorem :: MATRIX_4:49 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3"))))) "holds" (Bool (Set (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_3 :::"-"::: ) (Set "(" (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:50 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set "(" ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ")" ) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_3 :::"-"::: ) (Set (Var "M")))))) ; theorem :: MATRIX_4:51 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ))))) ; theorem :: MATRIX_4:52 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set "(" (Set (Var "M2")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" ) ")" ))))) ; theorem :: MATRIX_4:53 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set (Set (Var "M1")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M2")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M3")) ")" )))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:54 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "," (Set (Var "M3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M3")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M3")))) & (Bool (Set (Set (Var "M3")) ($#k1_matrix_4 :::"-"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "M3")) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set (Var "M2")) ")" )))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2"))))) ; theorem :: MATRIX_4:55 (Bool "for" (Set (Var "K")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B"))))) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "B")))))) ; theorem :: MATRIX_4:56 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "z"))))) "holds" (Bool (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set "(" (Set (Var "x")) ($#k3_fvsum_1 :::"+"::: ) (Set (Var "y")) ")" ) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set (Var "x")) "," (Set (Var "z")) ")" ")" ) ($#k3_fvsum_1 :::"+"::: ) (Set "(" ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set (Var "y")) "," (Set (Var "z")) ")" ")" ))))) ; theorem :: MATRIX_4:57 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "z"))))) "holds" (Bool (Set ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set (Var "z")) "," (Set "(" (Set (Var "x")) ($#k3_fvsum_1 :::"+"::: ) (Set (Var "y")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set (Var "z")) "," (Set (Var "x")) ")" ")" ) ($#k3_fvsum_1 :::"+"::: ) (Set "(" ($#k11_fvsum_1 :::"mlt"::: ) "(" (Set (Var "z")) "," (Set (Var "y")) ")" ")" ))))) ; theorem :: MATRIX_4:58 (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")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#m1_matrix_1 :::"Matrix"::: ) "of" (Set (Var "n")) "," (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) "," (Set (Var "D"))) "iff" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")))) ")" )))) ; theorem :: MATRIX_4:59 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B")))) & (Bool "ex" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "B")) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) ")" ")" ) ($#k3_fvsum_1 :::"+"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "B")) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: MATRIX_4:60 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B")))) & (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "B")) ")" ) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "A")) "," (Set (Var "j")) ")" ")" ) ($#k3_fvsum_1 :::"+"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "B")) "," (Set (Var "j")) ")" ")" )))))) ; theorem :: MATRIX_4:61 (Bool "for" (Set (Var "V1")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "P1")) "," (Set (Var "P2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "V1")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "P2"))))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "P1")) ($#k3_fvsum_1 :::"+"::: ) (Set (Var "P2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "P1")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "P2")) ")" ))))) ; theorem :: MATRIX_4:62 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "C")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "A")) ($#k4_matrix_3 :::"*"::: ) (Set "(" (Set (Var "B")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "B")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "C")) ")" ))))) ; theorem :: MATRIX_4:63 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "C")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "B")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "C")) ")" ) ($#k4_matrix_3 :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "B")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "A")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" (Set (Var "C")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "A")) ")" ))))) ; theorem :: MATRIX_4:64 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "M1")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "K")) (Bool "for" (Set (Var "M2")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "K")) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2")))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M1")))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M2"))))) "holds" (Bool (Set (Set (Var "M1")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "M2"))) "is" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "k")) "," (Set (Var "K"))))))) ;