let n be Nat; :: thesis: for K being Field
for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds
M1 @ is_congruent_Matrix_of M2 @
let K be Field; :: thesis: for M2, M1 being Matrix of n,K st M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 holds
M1 @ is_congruent_Matrix_of M2 @
let M2, M1 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 implies M1 @ is_congruent_Matrix_of M2 @ )
assume A1: ( M2 is invertible & M1 is_congruent_Matrix_of M2 & n > 0 ) ; :: thesis: M1 @ is_congruent_Matrix_of M2 @
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 @ ) * M2) * M4 ) by Def6;
A3: ( width M4 = n & len M4 = n ) by MATRIX_1:25;
A4: ( width M2 = n & len M2 = n ) by MATRIX_1:25;
A5: ( len (M4 @ ) = n & width (M4 @ ) = n ) by MATRIX_1:25;
A6: ( width (M2 * M4) = n & len (M2 * M4) = n ) by MATRIX_1:25;
A7: ((M4 @ ) * (M2 @ )) * M4 = ((M4 @ ) * (M2 @ )) * ((M4 @ ) @ ) by A1, A3, MATRIX_2:15
.= ((M2 * M4) @ ) * ((M4 @ ) @ ) by A1, A3, A4, MATRIX_3:24
.= ((M4 @ ) * (M2 * M4)) @ by A1, A5, A6, MATRIX_3:24
.= M1 @ by A2, A3, A4, A5, MATRIX_3:35 ;
take M4 ; :: according to MATRIX_8:def 6 :: thesis: ( M4 is invertible & M1 @ = ((M4 @ ) * (M2 @ )) * M4 )
thus ( M4 is invertible & M1 @ = ((M4 @ ) * (M2 @ )) * M4 ) by A2, A7; :: thesis: verum