:: MATRIX_9 semantic presentation begin theorem :: MATRIX_9:1 (Bool "for" (Set (Var "a")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "a")) ($#k1_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "A"))))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) "n" ")" )); end; scheme :: MATRIX_9:sch 1 NonEmptyFiniteX{ F1() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set F1 "(" ")" ) ")" )), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool P1[(Set F2 "(" ")" )]) provided (Bool "for" (Set (Var "x")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set F1 "(" ")" )) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool P1[(Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )])) and (Bool "for" (Set (Var "x")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set F1 "(" ")" )) (Bool "for" (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set F1 "(" ")" ) ")" )) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" )) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set F2 "(" ")" )) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) & (Bool P1[(Set (Var "B"))])) "holds" (Bool P1[(Set (Set (Var "B")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))]))) proof end; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n") ($#v4_relat_1 :::"-defined"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k4_relat_1 :::"id"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" )) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; theorem :: MATRIX_9:2 (Bool "(" (Bool (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Num 2) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Num 2) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ; theorem :: MATRIX_9:3 (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 2))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 2))) & (Bool (Bool "not" (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 2) ")" ))))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 2) ")" ) ")" )))) ; begin theorem :: MATRIX_9:4 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set (Var "n")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))))) ; theorem :: MATRIX_9:5 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")))))) ; theorem :: MATRIX_9:6 (Bool (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Num 2) ")" ) "," (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Num 2) ")" ) ")" ) ($#k2_tarski :::"}"::: ) )) ; begin definitionlet "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"PPath_product"::: "M" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) "n" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K") means :: MATRIX_9:def 1 (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) "n") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" "K") ($#k1_finsop_1 :::"$$"::: ) (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," "M" ")" ")" )))); end; :: deftheorem defines :::"PPath_product"::: MATRIX_9:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_matrix_9 :::"PPath_product"::: ) (Set (Var "M")))) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k1_finsop_1 :::"$$"::: ) (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ")" )))) ")" ))))); definitionlet "n" be ($#m1_hidden :::"Nat":::); let "K" be ($#l6_algstr_0 :::"Field":::); let "M" be ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Const "n")) "," (Set (Const "K")); func :::"Per"::: "M" -> ($#m1_subset_1 :::"Element":::) "of" "K" equals :: MATRIX_9:def 2 (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "K") ($#k7_setwiseo :::"$$"::: ) "(" (Set "(" ($#k15_matrix_2 :::"FinOmega"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) "n" ")" ) ")" ) "," (Set "(" ($#k1_matrix_9 :::"PPath_product"::: ) "M" ")" ) ")" ); end; :: deftheorem defines :::"Per"::: MATRIX_9:def 2 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "holds" (Bool (Set ($#k2_matrix_9 :::"Per"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "K"))) ($#k7_setwiseo :::"$$"::: ) "(" (Set "(" ($#k15_matrix_2 :::"FinOmega"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")) ")" ) ")" ) "," (Set "(" ($#k1_matrix_9 :::"PPath_product"::: ) (Set (Var "M")) ")" ) ")" ))))); theorem :: MATRIX_9:7 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k2_matrix_9 :::"Per"::: ) (Set ($#k5_matrix_2 :::"<*"::: ) (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "a")) ($#k4_matrix_2 :::"*>"::: ) ) ($#k5_matrix_2 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "a"))))) ; theorem :: MATRIX_9:8 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set ($#k2_matrix_9 :::"Per"::: ) (Set "(" ($#k1_matrix_3 :::"0."::: ) "(" (Set (Var "K")) "," (Set (Var "n")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))) ; theorem :: MATRIX_9:9 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_2 :::"idseq"::: ) (Num 2)))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "d")) ($#k2_finseq_4 :::"*>"::: ) ))))) ; theorem :: MATRIX_9:10 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Num 2) ")" )))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "b")) "," (Set (Var "c")) ($#k2_finseq_4 :::"*>"::: ) ))))) ; theorem :: MATRIX_9:11 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k1_finsop_1 :::"$$"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "b")))))) ; begin registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 2)) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 2)) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_funct_2 :::"onto"::: ) ($#v3_funct_2 :::"bijective"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_matrix_2 :::"odd"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 2) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 2) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n") ($#v4_relat_1 :::"-defined"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_funct_2 :::"onto"::: ) ($#v3_funct_2 :::"bijective"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_matrix_2 :::"even"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: MATRIX_9:12 (Bool (Set ($#k2_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) ($#k2_finseq_4 :::"*>"::: ) ) "is" ($#v5_matrix_2 :::"odd"::: ) ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 2) ")" )) ; theorem :: MATRIX_9:13 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "c")) ")" ))))) ; theorem :: MATRIX_9:14 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set ($#k2_matrix_9 :::"Per"::: ) (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "c")) ")" ))))) ; theorem :: MATRIX_9:15 (Bool (Set ($#k3_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Num 3) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ; theorem :: MATRIX_9:16 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "z")) "," (Set (Var "y")) "," (Set (Var "x")) ($#k3_finseq_4 :::"*>"::: ) ))))) ; theorem :: MATRIX_9:17 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "g"))) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set "(" ($#k3_finseq_5 :::"Rev"::: ) (Set (Var "g")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")))))) ; theorem :: MATRIX_9:18 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "g"))) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "g")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")))))) ; theorem :: MATRIX_9:19 (Bool (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k4_enumset1 :::"{"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k4_enumset1 :::"}"::: ) )) ; theorem :: MATRIX_9:20 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "e")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: MATRIX_9:21 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "e")) "," (Set (Var "g")) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: MATRIX_9:22 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "f")) "," (Set (Var "h")) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: MATRIX_9:23 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "b")) "," (Set (Var "f")) "," (Set (Var "g")) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: MATRIX_9:24 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "b")) "," (Set (Var "d")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: MATRIX_9:25 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "h")) ($#k3_finseq_4 :::"*>"::: ) )))))) ; theorem :: MATRIX_9:26 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k1_finsop_1 :::"$$"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "b")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "c")))))) ; theorem :: MATRIX_9:27 (Bool "(" (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3))) & (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3))) ")" ) ; theorem :: MATRIX_9:28 (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#k2_funct_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ; theorem :: MATRIX_9:29 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k13_matrix_2 :::"Group_of_Perm"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set (Set (Var "a")) ($#k2_group_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ))) ; begin theorem :: MATRIX_9:30 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) )) ; theorem :: MATRIX_9:31 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) )) ; theorem :: MATRIX_9:32 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) )) ; theorem :: MATRIX_9:33 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_funct_2 :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "not" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) )))) ; theorem :: MATRIX_9:34 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) ))) ; theorem :: MATRIX_9:35 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) ))) ; begin theorem :: MATRIX_9:36 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Var "f")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))))) ; theorem :: MATRIX_9:37 (Bool "(" (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) & (Bool (Set (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) ")" ) ; theorem :: MATRIX_9:38 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "p")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) ) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) ")" )) ; theorem :: MATRIX_9:39 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "holds" (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) ($#r2_hidden :::"in"::: ) (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")))))) ; theorem :: MATRIX_9:40 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "l")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k13_matrix_2 :::"Group_of_Perm"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "l")) ")" ) ($#k4_nat_d :::"mod"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "l"))))) "holds" (Bool "ex" (Set (Var "q")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool "(" (Bool (Set (Set (Var "l")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) & (Bool (Set (Var "q")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) ) ")" )) ")" )) "holds" (Bool (Set ($#k3_group_4 :::"Product"::: ) (Set (Var "l"))) "is" ($#v5_matrix_2 :::"even"::: ) ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) ; theorem :: MATRIX_9:41 (Bool "for" (Set (Var "l")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k13_matrix_2 :::"Group_of_Perm"::: ) (Num 3) ")" ) "st" (Bool (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "l")) ")" ) ($#k4_nat_d :::"mod"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "l"))))) "holds" (Bool "ex" (Set (Var "q")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Num 3)) "st" (Bool "(" (Bool (Set (Set (Var "l")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))) & (Bool (Set (Var "q")) "is" ($#v4_matrix_2 :::"being_transposition"::: ) ) ")" )) ")" ) & (Bool (Bool "not" (Set ($#k3_group_4 :::"Product"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 2) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ))) & (Bool (Bool "not" (Set ($#k3_group_4 :::"Product"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 3) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )))) "holds" (Bool (Set ($#k3_group_4 :::"Product"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 1) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ))) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 3)) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 3)) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_funct_1 :::"one-to-one"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_funct_2 :::"onto"::: ) ($#v3_funct_2 :::"bijective"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v5_matrix_2 :::"odd"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "," (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: MATRIX_9:42 (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) ) "is" ($#v5_matrix_2 :::"odd"::: ) ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" )) ; theorem :: MATRIX_9:43 (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) ) "is" ($#v5_matrix_2 :::"odd"::: ) ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" )) ; theorem :: MATRIX_9:44 (Bool (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) ) "is" ($#v5_matrix_2 :::"odd"::: ) ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" )) ; theorem :: MATRIX_9:45 (Bool "for" (Set (Var "p")) "being" ($#v5_matrix_2 :::"odd"::: ) ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Num 3) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 3) "," (Num 2) "," (Num 1) ($#k3_finseq_4 :::"*>"::: ) )) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 1) "," (Num 3) "," (Num 2) ($#k3_finseq_4 :::"*>"::: ) )) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Num 2) "," (Num 1) "," (Num 3) ($#k3_finseq_4 :::"*>"::: ) )) ")" )) ; begin theorem :: MATRIX_9:46 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "e")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "i")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "c")) ($#k8_group_1 :::"*"::: ) (Set (Var "e")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "h")) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "c")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "h")) ")" )))))) ; theorem :: MATRIX_9:47 (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Num 3) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k3_finseq_4 :::"*>"::: ) ) "," (Set ($#k3_finseq_4 :::"<*"::: ) (Set (Var "g")) "," (Set (Var "h")) "," (Set (Var "i")) ($#k3_finseq_4 :::"*>"::: ) ) ($#k3_finseq_4 :::"*>"::: ) ))) "holds" (Bool (Set ($#k2_matrix_9 :::"Per"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "e")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "i")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "c")) ($#k8_group_1 :::"*"::: ) (Set (Var "e")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "h")) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "g")) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "i")) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "c")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "h")) ")" )))))) ; theorem :: MATRIX_9:48 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" )))) ; theorem :: MATRIX_9:49 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" )))))) ; theorem :: MATRIX_9:50 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ))) "holds" (Bool (Set (Set "(" ($#k11_matrix_3 :::"Path_product"::: ) (Set (Var "M")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX_9:51 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ))) "holds" (Bool (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "K"))) ($#k7_setwiseo :::"$$"::: ) "(" (Set "(" ($#k15_matrix_2 :::"FinOmega"::: ) (Set "(" ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n")) ")" ) ")" ) "," (Set "(" ($#k11_matrix_3 :::"Path_product"::: ) (Set (Var "M")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))))) ; theorem :: MATRIX_9:52 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_matrix_9 :::"PPath_product"::: ) (Set (Var "M")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: MATRIX_9:53 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ))) "holds" (Bool (Set ($#k12_matrix_3 :::"Det"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))))) ; theorem :: MATRIX_9:54 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) ")" ) ")" ))) "holds" (Bool (Set ($#k2_matrix_9 :::"Per"::: ) (Set (Var "M"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))))) ; begin theorem :: MATRIX_9:55 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) & (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "N")) "," (Set (Var "i")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "N")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")))) ")" )))))) ; theorem :: MATRIX_9:56 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"Field":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m1_matrix_1 :::"Matrix":::) "of" (Set (Var "n")) "," (Set (Var "K")) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "N")) "," (Set (Var "i")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "l")) ($#r1_hidden :::"<>"::: ) (Set (Var "i"))) & (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "M")) "," (Set (Var "l")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_matrix_1 :::"Col"::: ) "(" (Set (Var "N")) "," (Set (Var "l")) ")" )) ")" ) ")" ))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_matrix_2 :::"Element"::: ) "of" (Set ($#k12_matrix_2 :::"Permutations"::: ) (Set (Var "n"))) (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "l")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "M")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" ($#k10_matrix_3 :::"Path_matrix"::: ) "(" (Set (Var "p")) "," (Set (Var "N")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "l")) ")" ))) ")" ))))))) ;