reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem
  for R being Ring, M1,M2 being Matrix of n,R
  holds M1 commutes_with M2 implies M1*M1 commutes_with M2
proof
  let R be Ring;
  let M1,M2 be Matrix of n,R;
A1: width M2=n by MATRIX_0:24;
A2: width M1=n & len M1=n by MATRIX_0:24;
  assume
A3: M1 commutes_with M2;
A4: len M2=n by MATRIX_0:24;
  then (M1*M1)*M2=M1*(M1*M2) by A2,MATRIX_3:33
    .=M1*(M2*M1) by A3
    .=(M1*M2)*M1 by A1,A2,A4,MATRIX_3:33
    .=(M2*M1)*M1 by A3
    .=M2*(M1*M1) by A1,A2,MATRIX_3:33;
  hence thesis;
end;
