let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is_congruent_Matrix_of M2 & n > 0 holds
(M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_congruent_Matrix_of M2 & n > 0 holds
(M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_congruent_Matrix_of M2 & n > 0 implies (M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2 )
assume A1: ( M1 is_congruent_Matrix_of M2 & n > 0 ) ; :: thesis: (M1 + M1) + M1 is_congruent_Matrix_of (M2 + M2) + M2
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 @ ) * M2) * M4 ) by Def6;
A3: ( len M4 = n & width M4 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A4: ( len (M2 + M2) = n & width (M2 + 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 + M2)) = n & width ((M4 @ ) * (M2 + M2)) = n ) by MATRIX_1:25;
A8: ((M4 @ ) * (M2 + M2)) * M4 = (((M4 @ ) * M2) + ((M4 @ ) * M2)) * M4 by A1, A3, A5, MATRIX_4:62
.= M1 + M1 by A1, A2, A3, A6, MATRIX_4:63 ;
A9: ((M4 @ ) * ((M2 + M2) + M2)) * M4 = (((M4 @ ) * (M2 + M2)) + ((M4 @ ) * M2)) * M4 by A1, A3, A4, A5, MATRIX_4:62
.= (M1 + M1) + M1 by A1, A2, A3, A6, A7, A8, MATRIX_4:63 ;
take M4 ; :: according to MATRIX_8:def 6 :: thesis: ( M4 is invertible & (M1 + M1) + M1 = ((M4 @ ) * ((M2 + M2) + M2)) * M4 )
thus ( M4 is invertible & (M1 + M1) + M1 = ((M4 @ ) * ((M2 + M2) + M2)) * M4 ) by A2, A9; :: thesis: verum