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
A2: ( M1 is Matrix of n,K & M2 is Matrix of n,K & width M1 = n & len M2 = n & width M2 = n & len M1 = n ) by MATRIX_1:25;
A3: ( M1 is invertible & M2 is invertible & M1 @ = M1 ~ & M2 @ = M2 ~ ) by A1, Def7;
then A4: ( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~ ) * (M1 ~ ) ) by Th46;
(M1 * M2) @ = (M2 ~ ) * (M1 ~ ) by A1, A2, A3, MATRIX_3:24;
hence M1 * M2 is Orthogonal by A4, Def7; :: thesis: verum