let n be Nat; :: thesis: for K being Field
for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds
M1 ~ is_similar_to M2 ~
let K be Field; :: thesis: for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds
M1 ~ is_similar_to M2 ~
let M2, M1 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_similar_to M2 implies M1 ~ is_similar_to M2 ~ )
assume A1: ( M2 is invertible & M1 is_similar_to M2 ) ; :: thesis: M1 ~ is_similar_to M2 ~
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 ~ ) * M2) * M4 ) by Def5;
A3: ( M4 ~ is invertible & (M4 ~ ) ~ = M4 ) by A2, MATRIX_6:16;
A4: ( M2 * M4 is invertible & (M4 ~ ) * (M2 ~ ) = (M2 * M4) ~ ) by A1, A2, MATRIX_6:37;
A5: ( len (M4 ~ ) = n & width (M4 ~ ) = n ) by MATRIX_1:25;
A6: ( len M4 = n & width M4 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A7: ((M4 ~ ) * (M2 ~ )) * M4 = ((M4 ~ ) * (M2 * M4)) ~ by A3, A4, MATRIX_6:37
.= M1 ~ by A2, A5, A6, MATRIX_3:35 ;
take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & M1 ~ = ((M4 ~ ) * (M2 ~ )) * M4 )
thus ( M4 is invertible & M1 ~ = ((M4 ~ ) * (M2 ~ )) * M4 ) by A2, A7; :: thesis: verum