let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is Orthogonal holds
M2 * M1 is_congruent_Matrix_of M1 * M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is Orthogonal holds
M2 * M1 is_congruent_Matrix_of M1 * M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is Orthogonal implies M2 * M1 is_congruent_Matrix_of M1 * M2 )
A1: ( len M2 = n & width (M1 ~ ) = n ) by MATRIX_1:25;
assume A2: M1 is Orthogonal ; :: thesis: M2 * M1 is_congruent_Matrix_of M1 * M2
then M1 is invertible by MATRIX_6:def 7;
then A3: M1 ~ is_reverse_of M1 by MATRIX_6:def 4;
take M1 ; :: according to MATRIX_8:def 6 :: thesis: ( M1 is invertible & M2 * M1 = ((M1 @ ) * (M1 * M2)) * M1 )
( len M1 = n & width M1 = n ) by MATRIX_1:25;
then ((M1 ~ ) * (M1 * M2)) * M1 = (((M1 ~ ) * M1) * M2) * M1 by A1, MATRIX_3:35
.= ((1. K,n) * M2) * M1 by A3, MATRIX_6:def 2
.= M2 * M1 by MATRIX_3:20 ;
hence ( M1 is invertible & M2 * M1 = ((M1 @ ) * (M1 * M2)) * M1 ) by A2, MATRIX_6:def 7; :: thesis: verum