:: MATRIXR1 semantic presentation begin theorem :: MATRIXR1:1 (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "fr1")) "," (Set (Var "fr2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set (Var "r1")) ($#r1_hidden :::"="::: ) (Set (Var "fr1"))) & (Bool (Set (Var "r2")) ($#r1_hidden :::"="::: ) (Set (Var "fr2")))) "holds" (Bool (Set (Set (Var "r1")) ($#k7_real_1 :::"+"::: ) (Set (Var "r2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "fr1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "fr2")))))) ; theorem :: MATRIXR1:2 (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "fr1")) "," (Set (Var "fr2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set (Var "r1")) ($#r1_hidden :::"="::: ) (Set (Var "fr1"))) & (Bool (Set (Var "r2")) ($#r1_hidden :::"="::: ) (Set (Var "fr2")))) "holds" (Bool (Set (Set (Var "r1")) ($#k8_real_1 :::"*"::: ) (Set (Var "r2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "fr1")) ($#k8_group_1 :::"*"::: ) (Set (Var "fr2")))))) ; theorem :: MATRIXR1:3 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "F")) ($#k4_rvsum_1 :::"+"::: ) (Set "(" ($#k6_rvsum_1 :::"-"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")) ")" ))) & (Bool (Set (Set (Var "F")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")) ")" ))) ")" )) ; theorem :: MATRIXR1:4 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))))) "holds" (Bool (Set (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "F1")) ($#k4_rvsum_1 :::"+"::: ) (Set "(" ($#k6_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" )))) ; theorem :: MATRIXR1:5 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "F")) ($#k8_rvsum_1 :::"-"::: ) (Set "(" ($#k5_euclid :::"0*"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "F")))) ; theorem :: MATRIXR1:6 (Bool "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_euclid :::"0*"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "F")) ")" ) ")" ) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k6_rvsum_1 :::"-"::: ) (Set (Var "F"))))) ; theorem :: MATRIXR1:7 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))))) "holds" (Bool (Set (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set "(" ($#k6_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F2"))))) ; theorem :: MATRIXR1:8 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))))) "holds" (Bool (Set ($#k6_rvsum_1 :::"-"::: ) (Set "(" (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "F2")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F1"))))) ; theorem :: MATRIXR1:9 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))))) "holds" (Bool (Set ($#k6_rvsum_1 :::"-"::: ) (Set "(" (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_rvsum_1 :::"-"::: ) (Set (Var "F1")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F2"))))) ; theorem :: MATRIXR1:10 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2")))) & (Bool (Set (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1")) ")" )))) "holds" (Bool (Set (Var "F1")) ($#r1_hidden :::"="::: ) (Set (Var "F2")))) ; theorem :: MATRIXR1:11 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "," (Set (Var "F3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F3"))))) "holds" (Bool (Set (Set "(" (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" ) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F3"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set "(" (Set (Var "F2")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F3")) ")" )))) ; theorem :: MATRIXR1:12 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "," (Set (Var "F3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F3"))))) "holds" (Bool (Set (Set (Var "F1")) ($#k4_rvsum_1 :::"+"::: ) (Set "(" (Set (Var "F2")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F2")) ")" ) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F3"))))) ; theorem :: MATRIXR1:13 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "," (Set (Var "F3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F3"))))) "holds" (Bool (Set (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set "(" (Set (Var "F2")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F3")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F3"))))) ; theorem :: MATRIXR1:14 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))))) "holds" (Bool (Set (Var "F1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F2")) ")" ) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2"))))) ; theorem :: MATRIXR1:15 (Bool "for" (Set (Var "F1")) "," (Set (Var "F2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F2"))))) "holds" (Bool (Set (Var "F1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "F1")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "F2")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "F2"))))) ; begin theorem :: MATRIXR1:16 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))))) ; theorem :: MATRIXR1:17 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "fr")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Var "fr"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "fp")))) "holds" (Bool (Set (Set (Var "r")) ($#k10_rvsum_1 :::"*"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "fr")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "fp")))))))) ; theorem :: MATRIXR1:18 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))))))) ; theorem :: MATRIXR1:19 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: MATRIXR1:20 (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" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_fvsum_1 :::"*"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: MATRIXR1:21 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))))) "holds" (Bool "ex" (Set (Var "C")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B")))) & (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "C"))))) "holds" (Bool (Set (Set (Var "C")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) ")" ")" ) ($#k13_fvsum_1 :::""*""::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "B")) "," (Set (Var "j")) ")" ")" ))) ")" ) ")" )))) ; theorem :: MATRIXR1:22 (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 "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Set (Var "A")) ($#k4_matrix_3 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k6_matrix_3 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_3 :::"*"::: ) (Set (Var "B")) ")" )))))) ; definitionlet "A" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func :::"MXR2MXF"::: "A" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) equals :: MATRIXR1:def 1 "A"; end; :: deftheorem defines :::"MXR2MXF"::: MATRIXR1:def 1 : (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))); definitionlet "A" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ); func :::"MXF2MXR"::: "A" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 2 "A"; end; :: deftheorem defines :::"MXF2MXR"::: MATRIXR1:def 2 : (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "holds" (Bool (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))); theorem :: MATRIXR1:23 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D1")) (Bool "for" (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D2")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#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 "A"))))) "holds" (Bool (Set (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )))))) ; theorem :: MATRIXR1:24 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" (Set (Var "A")) ($#k3_matrix_3 :::"+"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))))) ; definitionlet "A", "B" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func "A" :::"+"::: "B" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 3 (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "A" ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "B" ")" ) ")" )); end; :: deftheorem defines :::"+"::: MATRIXR1:def 3 : (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "A")) ")" ) ($#k3_matrix_3 :::"+"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "B")) ")" ) ")" )))); theorem :: MATRIXR1:25 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")))) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#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 "A"))))) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "B")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ))) ")" ) ")" )) ; theorem :: MATRIXR1:26 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")))) & (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 ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "C")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "B")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ))) ")" )) "holds" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B"))))) ; definitionlet "A" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func :::"-"::: "A" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 4 (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "A" ")" ) ")" )); end; :: deftheorem defines :::"-"::: MATRIXR1:def 4 : (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k4_matrixr1 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" ($#k2_matrix_3 :::"-"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "A")) ")" ) ")" )))); definitionlet "A", "B" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func "A" :::"-"::: "B" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 5 (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "A" ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "B" ")" ) ")" )); func "A" :::"*"::: "B" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 6 (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "A" ")" ) ($#k4_matrix_3 :::"*"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "B" ")" ) ")" )); end; :: deftheorem defines :::"-"::: MATRIXR1:def 5 : (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k5_matrixr1 :::"-"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "A")) ")" ) ($#k1_matrix_4 :::"-"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "B")) ")" ) ")" )))); :: deftheorem defines :::"*"::: MATRIXR1:def 6 : (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "A")) ")" ) ($#k4_matrix_3 :::"*"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "B")) ")" ) ")" )))); definitionlet "a" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "A" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func "a" :::"*"::: "A" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: MATRIXR1:def 7 (Bool "for" (Set (Var "ea")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set (Var "ea")) ($#r1_hidden :::"="::: ) "a")) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set (Var "ea")) ($#k6_matrix_3 :::"*"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) "A" ")" ) ")" )))); end; :: deftheorem defines :::"*"::: MATRIXR1:def 7 : (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "b3")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")))) "iff" (Bool "for" (Set (Var "ea")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "st" (Bool (Bool (Set (Var "ea")) ($#r1_hidden :::"="::: ) (Set (Var "a")))) "holds" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" (Set (Var "ea")) ($#k6_matrix_3 :::"*"::: ) (Set "(" ($#k1_matrixr1 :::"MXR2MXF"::: ) (Set (Var "A")) ")" ) ")" )))) ")" ))); theorem :: MATRIXR1:27 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")))) ")" ))) ; theorem :: MATRIXR1:28 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A")))))) ; theorem :: MATRIXR1:29 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i2")) "," (Set (Var "j2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" )))))) ; theorem :: MATRIXR1:30 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ))))) ; theorem :: MATRIXR1:31 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "A"))))) "holds" (Bool "(" (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) & (Bool "(" "for" (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set (Var "q")))) ")" ) ")" )))) ; theorem :: MATRIXR1:32 (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Num 1) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))) ; theorem :: MATRIXR1:33 (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A"))))) ; theorem :: MATRIXR1:34 (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "A")) ")" ) ($#k3_matrixr1 :::"+"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Num 3) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A"))))) ; definitionlet "n", "m" be ($#m1_hidden :::"Nat":::); func :::"0_Rmatrix"::: "(" "n" "," "m" ")" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 8 (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" ($#k1_matrix_3 :::"0."::: ) "(" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "," "n" "," "m" ")" ")" )); end; :: deftheorem defines :::"0_Rmatrix"::: MATRIXR1:def 8 : (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set (Var "n")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_matrixr1 :::"MXF2MXR"::: ) (Set "(" ($#k1_matrix_3 :::"0."::: ) "(" (Set ($#k2_vectsp_1 :::"F_Real"::: ) ) "," (Set (Var "n")) "," (Set (Var "m")) ")" ")" )))); theorem :: MATRIXR1:35 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k5_matrixr1 :::"-"::: ) (Set "(" ($#k4_matrixr1 :::"-"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B"))))) ; theorem :: MATRIXR1:36 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set "(" ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set (Var "n")) "," (Set (Var "m")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Set "(" ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set (Var "n")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrixr1 :::"+"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" ))) ; theorem :: MATRIXR1:37 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#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 ($#k1_matrix_1 :::"width"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ))) ; theorem :: MATRIXR1:38 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#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 ($#k1_matrix_1 :::"width"::: ) (Set (Var "B")))) & (Bool (Set (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ))) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k4_matrixr1 :::"-"::: ) (Set (Var "A"))))) ; theorem :: MATRIXR1:39 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "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")))) & (Bool (Set (Set (Var "B")) ($#k5_matrixr1 :::"-"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ))) ; theorem :: MATRIXR1:40 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))))) "holds" (Bool (Set (Set (Var "A")) ($#k6_matrixr1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "B")) ")" ))))) ; theorem :: MATRIXR1:41 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (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 ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k6_matrixr1 :::"*"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k6_matrixr1 :::"*"::: ) (Set (Var "B")) ")" ))))) ; theorem :: MATRIXR1:42 (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "M")) ($#k3_matrixr1 :::"+"::: ) (Set "(" ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "M")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "M")) ")" ) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "M")))) ; theorem :: MATRIXR1:43 (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "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 (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k3_matrixr1 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "B")) ")" ))))) ; theorem :: MATRIXR1:44 (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set ($#k6_numbers :::"0"::: ) ) ($#k7_matrixr1 :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")) ")" ) ")" ))) ; definitionlet "x" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); assume (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Const "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ; func :::"ColVec2Mx"::: "x" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: MATRIXR1:def 9 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "x")) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) it) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "x"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" "x" ($#k1_seq_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"ColVec2Mx"::: MATRIXR1:def 9 : (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "x")) ($#k1_seq_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ")" ) ")" ) ")" ))); theorem :: MATRIXR1:45 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ))) ; theorem :: MATRIXR1:46 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set "(" (Set (Var "x1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x1")) ")" ) ($#k3_matrixr1 :::"+"::: ) (Set "(" ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x2")) ")" )))) ; theorem :: MATRIXR1:47 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set "(" (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set "(" ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x")) ")" ))))) ; definitionlet "x" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func :::"LineVec2Mx"::: "x" -> ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) means :: MATRIXR1:def 10 (Bool "(" (Bool (Set ($#k1_matrix_1 :::"width"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "x")) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "x"))) "holds" (Bool (Set it ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set "x" ($#k1_seq_1 :::"."::: ) (Set (Var "j")))) ")" ) ")" ); end; :: deftheorem defines :::"LineVec2Mx"::: MATRIXR1:def 10 : (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k1_seq_1 :::"."::: ) (Set (Var "j")))) ")" ) ")" ) ")" ))); theorem :: MATRIXR1:48 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "M")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" ))) ; theorem :: MATRIXR1:49 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x")) ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x")))) & (Bool (Set (Set "(" ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x")) ")" ) ($#k4_matrix_1 :::"@"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x")))) ")" )) ; theorem :: MATRIXR1:50 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x2"))))) "holds" (Bool (Set ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set "(" (Set (Var "x1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x1")) ")" ) ($#k3_matrixr1 :::"+"::: ) (Set "(" ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x2")) ")" )))) ; theorem :: MATRIXR1:51 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set "(" (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set "(" ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x")) ")" ))))) ; definitionlet "M" be ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ); let "x" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); func "M" :::"*"::: "x" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 11 (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" "M" ($#k6_matrixr1 :::"*"::: ) (Set "(" ($#k9_matrixr1 :::"ColVec2Mx"::: ) "x" ")" ) ")" ) "," (Num 1) ")" ); func "x" :::"*"::: "M" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) equals :: MATRIXR1:def 12 (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set "(" ($#k10_matrixr1 :::"LineVec2Mx"::: ) "x" ")" ) ($#k6_matrixr1 :::"*"::: ) "M" ")" ) "," (Num 1) ")" ); end; :: deftheorem defines :::"*"::: MATRIXR1:def 11 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "M")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" (Set (Var "M")) ($#k6_matrixr1 :::"*"::: ) (Set "(" ($#k9_matrixr1 :::"ColVec2Mx"::: ) (Set (Var "x")) ")" ) ")" ) "," (Num 1) ")" )))); :: deftheorem defines :::"*"::: MATRIXR1:def 12 : (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set "(" ($#k10_matrixr1 :::"LineVec2Mx"::: ) (Set (Var "x")) ")" ) ($#k6_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) "," (Num 1) ")" )))); theorem :: MATRIXR1:52 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) "or" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A")))))) ; theorem :: MATRIXR1:53 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) "or" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) ")" )) "holds" (Bool (Set (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k4_matrix_1 :::"@"::: ) ")" ))))) ; theorem :: MATRIXR1:54 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "B"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) ")" ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "B")) "," (Set (Var "i")) ")" ")" ))))) ; theorem :: MATRIXR1:55 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "B"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))))) "holds" (Bool (Set ($#k8_matrix_1 :::"Line"::: ) "(" (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "A")) "," (Set (Var "i")) ")" ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" ($#k8_matrix_1 :::"Line"::: ) "(" (Set (Var "B")) "," (Set (Var "i")) ")" ")" ))))) ; theorem :: MATRIXR1:56 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "M"))))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set "(" (Set (Var "a")) ($#k7_matrixr1 :::"*"::: ) (Set (Var "M")) ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k11_rvsum_1 :::"*"::: ) (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" )))))) ; theorem :: MATRIXR1:57 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x2")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set "(" (Set (Var "x1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x1")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x2")) ")" ))))) ; theorem :: MATRIXR1:58 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x2")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x1"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "x1")) ($#k4_rvsum_1 :::"+"::: ) (Set (Var "x2")) ")" ) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x1")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" (Set (Var "x2")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ))))) ; theorem :: MATRIXR1:59 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" )))))) ; theorem :: MATRIXR1:60 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A")) ")" )))))) ; theorem :: MATRIXR1:61 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A")))))) ; theorem :: MATRIXR1:62 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A")))))) ; theorem :: MATRIXR1:63 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "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")))) & (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" ) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" (Set (Var "B")) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x")) ")" ))))) ; theorem :: MATRIXR1:64 (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#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 ($#k1_matrix_1 :::"width"::: ) (Set (Var "B")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set "(" (Set (Var "A")) ($#k3_matrixr1 :::"+"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "A")) ")" ) ($#k4_rvsum_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set (Var "B")) ")" ))))) ; theorem :: MATRIXR1:65 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set (Var "n")) "," (Set (Var "m")) ")" ")" ) ($#k11_matrixr1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "n")))))) ; theorem :: MATRIXR1:66 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "x")) ($#k12_matrixr1 :::"*"::: ) (Set "(" ($#k8_matrixr1 :::"0_Rmatrix"::: ) "(" (Set (Var "n")) "," (Set (Var "m")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "m")))))) ;