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 commutative Ring, M1,M2 being Matrix of n,R
  holds n>0 & M1 commutes_with M2 implies M1@ commutes_with M2@
proof
  let R be commutative Ring;
  let M1,M2 be Matrix of n,R;
A1: width M1=n & width M2=n by MATRIX_0:24;
  set M3=M1@, M4=M2@;
A2: len M2=n by MATRIX_0:24;
  assume that
A3: n>0 and
A4: M1 commutes_with M2;
  len M1=n by MATRIX_0:24;
  then M3*M4=(M2*M1)@ by A1,A3,MATRIX_3:22
    .=(M1*M2)@ by A4
    .=M4*M3 by A1,A2,A3,MATRIX_3:22;
  hence thesis;
end;
