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