let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & n > 0 holds
M1 + M1 is_similar_to M2 + M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & n > 0 holds
M1 + M1 is_similar_to M2 + M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 & n > 0 implies M1 + M1 is_similar_to M2 + M2 )
assume A1: ( M1 is_similar_to M2 & n > 0 ) ; :: thesis: M1 + M1 is_similar_to M2 + M2
then consider M4 being Matrix of n,K such that
A2: ( M4 is invertible & M1 = ((M4 ~ ) * M2) * M4 ) by Def5;
A3: ( len M4 = n & width M4 = n & len M2 = n & width M2 = n ) by MATRIX_1:25;
A4: ( len (M4 ~ ) = n & width (M4 ~ ) = n ) by MATRIX_1:25;
A5: ( len ((M4 ~ ) * M2) = n & width ((M4 ~ ) * M2) = n ) by MATRIX_1:25;
A6: ((M4 ~ ) * (M2 + M2)) * M4 = (((M4 ~ ) * M2) + ((M4 ~ ) * M2)) * M4 by A1, A3, A4, MATRIX_4:62
.= M1 + M1 by A1, A2, A3, A5, MATRIX_4:63 ;
take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & M1 + M1 = ((M4 ~ ) * (M2 + M2)) * M4 )
thus ( M4 is invertible & M1 + M1 = ((M4 ~ ) * (M2 + M2)) * M4 ) by A2, A6; :: thesis: verum