let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds
M1 * M1 commutes_with M2
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds
M1 * M1 commutes_with M2
let M1, M2 be Matrix of n,K; :: thesis: ( M1 commutes_with M2 implies M1 * M1 commutes_with M2 )
A1: ( width M1 = n & width M2 = n & len M1 = n & len M2 = n ) by MATRIX_1:25;
assume A2: M1 commutes_with M2 ; :: thesis: M1 * M1 commutes_with M2
(M1 * M1) * M2 = M1 * (M1 * M2) by A1, MATRIX_3:35
.= M1 * (M2 * M1) by A2, Def1
.= (M1 * M2) * M1 by A1, MATRIX_3:35
.= (M2 * M1) * M1 by A2, Def1
.= M2 * (M1 * M1) by A1, MATRIX_3:35 ;
hence M1 * M1 commutes_with M2 by Def1; :: thesis: verum