let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & M2 is Involutory holds
M1 is Involutory
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & M2 is Involutory holds
M1 is Involutory
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 & M2 is Involutory implies M1 is Involutory )
assume A1: ( M1 is_similar_to M2 & M2 is Involutory ) ; :: thesis: M1 is Involutory
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 ~ ) * M2) * M4 ) by Def5;
A3: ( len (M2 * M4) = n & width (M2 * M4) = n ) by MATRIX_1:25;
A4: ( len M4 = n & width M4 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A5: ( len (M4 ~ ) = n & width (M4 ~ ) = n ) by MATRIX_1:25;
A6: ( len ((M4 ~ ) * M2) = n & width ((M4 ~ ) * M2) = n ) by MATRIX_1:25;
A7: ( len (((M4 ~ ) * M2) * M4) = n & width (((M4 ~ ) * M2) * M4) = n ) by MATRIX_1:25;
A8: M4 ~ is_reverse_of M4 by A2, MATRIX_6:def 4;
M1 * M1 = (((M4 ~ ) * M2) * M4) * ((M4 ~ ) * (M2 * M4)) by A2, A4, A5, MATRIX_3:35
.= ((((M4 ~ ) * M2) * M4) * (M4 ~ )) * (M2 * M4) by A3, A5, A7, MATRIX_3:35
.= (((M4 ~ ) * M2) * (M4 * (M4 ~ ))) * (M2 * M4) by A4, A5, A6, MATRIX_3:35
.= (((M4 ~ ) * M2) * (1. K,n)) * (M2 * M4) by A8, MATRIX_6:def 2
.= ((M4 ~ ) * M2) * (M2 * M4) by MATRIX_3:21
.= (((M4 ~ ) * M2) * M2) * M4 by A4, A6, MATRIX_3:35
.= ((M4 ~ ) * (M2 * M2)) * M4 by A4, A5, MATRIX_3:35
.= ((M4 ~ ) * (1. K,n)) * M4 by A1, Def3
.= (M4 ~ ) * M4 by MATRIX_3:21
.= 1. K,n by A8, MATRIX_6:def 2 ;
hence M1 is Involutory by Def3; :: thesis: verum