let K be Field; :: thesis: for a, b, c, d being Element of K

for p being Element of Permutations 2 st p = Rev (idseq 2) holds

Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*>

let a, b, c, d be Element of K; :: thesis: for p being Element of Permutations 2 st p = Rev (idseq 2) holds

Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*>

let p be Element of Permutations 2; :: thesis: ( p = Rev (idseq 2) implies Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*> )

assume A1: p = Rev (idseq 2) ; :: thesis: Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*>

A2: len (Path_matrix (p,((a,b) ][ (c,d)))) = 2 by MATRIX_3:def 7;

then A3: dom (Path_matrix (p,((a,b) ][ (c,d)))) = Seg 2 by FINSEQ_1:def 3;

then 1 in dom (Path_matrix (p,((a,b) ][ (c,d)))) ;

then A4: (Path_matrix (p,((a,b) ][ (c,d)))) . 1 = ((a,b) ][ (c,d)) * (1,2) by A1, Th2, MATRIX_3:def 7

.= b by MATRIX_0:50 ;

2 in dom (Path_matrix (p,((a,b) ][ (c,d)))) by A3;

then (Path_matrix (p,((a,b) ][ (c,d)))) . 2 = ((a,b) ][ (c,d)) * (2,1) by A1, Th2, MATRIX_3:def 7

.= c by MATRIX_0:50 ;

hence Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*> by A2, A4, FINSEQ_1:44; :: thesis: verum

for p being Element of Permutations 2 st p = Rev (idseq 2) holds

Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*>

let a, b, c, d be Element of K; :: thesis: for p being Element of Permutations 2 st p = Rev (idseq 2) holds

Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*>

let p be Element of Permutations 2; :: thesis: ( p = Rev (idseq 2) implies Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*> )

assume A1: p = Rev (idseq 2) ; :: thesis: Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*>

A2: len (Path_matrix (p,((a,b) ][ (c,d)))) = 2 by MATRIX_3:def 7;

then A3: dom (Path_matrix (p,((a,b) ][ (c,d)))) = Seg 2 by FINSEQ_1:def 3;

then 1 in dom (Path_matrix (p,((a,b) ][ (c,d)))) ;

then A4: (Path_matrix (p,((a,b) ][ (c,d)))) . 1 = ((a,b) ][ (c,d)) * (1,2) by A1, Th2, MATRIX_3:def 7

.= b by MATRIX_0:50 ;

2 in dom (Path_matrix (p,((a,b) ][ (c,d)))) by A3;

then (Path_matrix (p,((a,b) ][ (c,d)))) . 2 = ((a,b) ][ (c,d)) * (2,1) by A1, Th2, MATRIX_3:def 7

.= c by MATRIX_0:50 ;

hence Path_matrix (p,((a,b) ][ (c,d))) = <*b,c*> by A2, A4, FINSEQ_1:44; :: thesis: verum