let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is_similar_to M2 holds
(M1 + M1) + M1 is_similar_to (M2 + M2) + M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 holds
(M1 + M1) + M1 is_similar_to (M2 + M2) + M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 implies (M1 + M1) + M1 is_similar_to (M2 + M2) + M2 )
assume M1 is_similar_to M2 ; :: thesis: (M1 + M1) + M1 is_similar_to (M2 + M2) + M2
then consider M4 being Matrix of n,K such that
A3: M4 is invertible and
A4: M1 = ((M4 ~) * M2) * M4 ;
A5: ( len M4 = n & len ((M4 ~) * M2) = n ) by MATRIX_0:24;
A6: width ((M4 ~) * M2) = n by MATRIX_0:24;
A7: ( len M2 = n & width M2 = n ) by MATRIX_0:24;
A8: ( len (M4 ~) = n & width (M4 ~) = n ) by MATRIX_0:24;
then A9: ((M4 ~) * (M2 + M2)) * M4 = (((M4 ~) * M2) + ((M4 ~) * M2)) * M4 by A7, MATRIX_4:62
.= M1 + M1 by A4, A5, A6, MATRIX_4:63 ;
take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & (M1 + M1) + M1 = ((M4 ~) * ((M2 + M2) + M2)) * M4 )
A10: ( len ((M4 ~) * (M2 + M2)) = n & width ((M4 ~) * (M2 + M2)) = n ) by MATRIX_0:24;
( len (M2 + M2) = n & width (M2 + M2) = n ) by MATRIX_0:24;
then ((M4 ~) * ((M2 + M2) + M2)) * M4 = (((M4 ~) * (M2 + M2)) + ((M4 ~) * M2)) * M4 by A7, A8, MATRIX_4:62
.= (M1 + M1) + M1 by A4, A5, A6, A10, A9, MATRIX_4:63 ;
hence ( M4 is invertible & (M1 + M1) + M1 = ((M4 ~) * ((M2 + M2) + M2)) * M4 ) by A3; :: thesis: verum