let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st n > 0 & M1 is Orthogonal & M2 is Orthogonal holds
(M1 ~ ) * M2 is Orthogonal
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 is Orthogonal & M2 is Orthogonal holds
(M1 ~ ) * M2 is Orthogonal
let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 is Orthogonal & M2 is Orthogonal implies (M1 ~ ) * M2 is Orthogonal )
assume A1: ( n > 0 & M1 is Orthogonal & M2 is Orthogonal ) ; :: thesis: (M1 ~ ) * M2 is Orthogonal
then A2: ( M1 is invertible & M2 is invertible & M1 @ = M1 ~ & M2 @ = M2 ~ ) by Def7;
A3: ( width M1 = n & len M1 = n & width (M1 ~ ) = n & width M2 = n & len (M1 ~ ) = n & len M2 = n ) by MATRIX_1:25;
M1 ~ is invertible by A2, Th16;
then A4: ( (M1 ~ ) * M2 is invertible & ((M1 ~ ) * M2) ~ = (M2 ~ ) * ((M1 ~ ) ~ ) ) by A2, Th37;
then A5: ((M1 ~ ) * M2) ~ = (M2 ~ ) * M1 by A2, Th16;
((M1 ~ ) * M2) @ = (M2 ~ ) * ((M1 @ ) @ ) by A1, A2, A3, MATRIX_3:24
.= (M2 ~ ) * M1 by A1, A3, MATRIX_2:15 ;
hence (M1 ~ ) * M2 is Orthogonal by A4, A5, Def7; :: thesis: verum