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
  for M1,M2,M3 being Matrix of n,R
  holds
  M1 commutes_with M2 & M1 commutes_with M3 implies
    M1 commutes_with M2+M3
proof
  let R be Ring;
  let M1,M2,M3 be Matrix of n,R;
A1: width M1=n by MATRIX_0:24;
A2: len M1=n & len M2=n by MATRIX_0:24;
A3: len M3=n by MATRIX_0:24;
  assume that
A4: M1 commutes_with M2 and
A5: M1 commutes_with M3;
A6: width M2=n & width M3=n by MATRIX_0:24;
  then (M2 + M3) * M1 = (M2 * M1) + (M3 * M1) by A2,A3,MATRIX_4:63
    .= M1*M2 + M3*M1 by A4
    .= M1*M2 + M1*M3 by A5
    .= M1*(M2 + M3) by A1,A6,A2,A3,MATRIX_4:62;
  hence thesis;
end;
