let n be Nat; :: thesis: for R being Ring
for M1, M2 being Matrix of n,R st M1 commutes_with M2 holds
M1 + M2 commutes_with M2 + M2
let R be Ring; :: thesis: for M1, M2 being Matrix of n,R st M1 commutes_with M2 holds
M1 + M2 commutes_with M2 + M2
let M1, M2 be Matrix of n,R; :: thesis: ( M1 commutes_with M2 implies M1 + M2 commutes_with M2 + M2 )
assume A1: M1 commutes_with M2 ; :: thesis: M1 + M2 commutes_with M2 + M2
A2: ( len M2 = n & width M2 = n ) by MATRIX_0:24;
A3: len (M1 + M2) = n by MATRIX_0:24;
A4: ( width M1 = n & len M1 = n ) by MATRIX_0:24;
width (M1 + M2) = n by MATRIX_0:24;
then (M1 + M2) * (M2 + M2) = ((M1 + M2) * M2) + ((M1 + M2) * M2) by A2, MATRIX_4:62
.= ((M1 * M2) + (M2 * M2)) + ((M1 + M2) * M2) by A2, A4, MATRIX_4:63
.= ((M1 * M2) + (M2 * M2)) + ((M1 * M2) + (M2 * M2)) by A2, A4, MATRIX_4:63
.= ((M2 * M1) + (M2 * M2)) + ((M1 * M2) + (M2 * M2)) by A1
.= ((M2 * M1) + (M2 * M2)) + ((M2 * M1) + (M2 * M2)) by A1
.= (M2 * (M1 + M2)) + ((M2 * M1) + (M2 * M2)) by A2, A4, MATRIX_4:62
.= (M2 * (M1 + M2)) + (M2 * (M1 + M2)) by A2, A4, MATRIX_4:62
.= (M2 + M2) * (M1 + M2) by A2, A3, MATRIX_4:63 ;
hence M1 + M2 commutes_with M2 + M2 ; :: thesis: verum