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_2:6 ;
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_2:6 ;
hence Path_matrix p,(a,b ][ c,d) = <*b,c*> by A2, A4, FINSEQ_1:61; :: thesis: verum