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;
  assume
A1: M1 commutes_with M2;
A2: width M2=n by MATRIX_0:24;
A3: len M1=n by MATRIX_0:24;
A4: width M1=n & len M2=n by MATRIX_0:24;
  then (M1+M1)*M2=M1*M2+M1*M2 by A3,MATRIX_4:63
    .=M2*M1+M1*M2 by A1
    .=M2*M1+M2*M1 by A1
    .=M2*(M1+M1) by A4,A2,A3,MATRIX_4:62;
  hence thesis;
end;
