let n be Nat; :: thesis: for K being commutative Ring
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 commutative Ring; :: 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 that
A1: n > 0 and
A2: ( M1 is Orthogonal & M2 is Orthogonal ) ; :: thesis: M1 * M2 is Orthogonal
( M1 is invertible & M2 is invertible ) by A2;
then A3: ( M1 * M2 is invertible & (M1 * M2) ~ = (M2 ~) * (M1 ~) ) by Th46;
A4: width M2 = n by MATRIX_0:24;
A5: ( width M1 = n & len M2 = n ) by MATRIX_0:24;
( M1 @ = M1 ~ & M2 @ = M2 ~ ) by A2;
then (M1 * M2) @ = (M2 ~) * (M1 ~) by A1, A5, A4, MATRIX_3:22;
hence M1 * M2 is Orthogonal by A3; :: thesis: verum