let n be Nat; :: thesis: for K being Field

for M2, M4 being Matrix of n,K st M4 is Orthogonal holds

((M4 @) * M2) * M4 is_similar_to M2

let K be Field; :: thesis: for M2, M4 being Matrix of n,K st M4 is Orthogonal holds

((M4 @) * M2) * M4 is_similar_to M2

let M2, M4 be Matrix of n,K; :: thesis: ( M4 is Orthogonal implies ((M4 @) * M2) * M4 is_similar_to M2 )

assume A1: M4 is Orthogonal ; :: thesis: ((M4 @) * M2) * M4 is_similar_to M2

take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & ((M4 @) * M2) * M4 = ((M4 ~) * M2) * M4 )

thus ( M4 is invertible & ((M4 @) * M2) * M4 = ((M4 ~) * M2) * M4 ) by A1, MATRIX_6:def 7; :: thesis: verum

for M2, M4 being Matrix of n,K st M4 is Orthogonal holds

((M4 @) * M2) * M4 is_similar_to M2

let K be Field; :: thesis: for M2, M4 being Matrix of n,K st M4 is Orthogonal holds

((M4 @) * M2) * M4 is_similar_to M2

let M2, M4 be Matrix of n,K; :: thesis: ( M4 is Orthogonal implies ((M4 @) * M2) * M4 is_similar_to M2 )

assume A1: M4 is Orthogonal ; :: thesis: ((M4 @) * M2) * M4 is_similar_to M2

take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & ((M4 @) * M2) * M4 = ((M4 ~) * M2) * M4 )

thus ( M4 is invertible & ((M4 @) * M2) * M4 = ((M4 ~) * M2) * M4 ) by A1, MATRIX_6:def 7; :: thesis: verum