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 is symmetric & M2 is symmetric implies 
  (M1 commutes_with M2 iff M1*M2 is symmetric)
proof
  let R be commutative Ring;
  let M1,M2 be Matrix of n,R;
  assume that
A1: n > 0 and
A2: M1 is symmetric & M2 is symmetric;
A3: width M1=n & len M2=n by MATRIX_0:24;
A4: width M2=n by MATRIX_0:24;
A5: M1@=M1 & M2@=M2 by A2;
  thus M1 commutes_with M2 implies M1*M2 is symmetric
  by A1,A5,A3,A4,MATRIX_3:22;
  assume
A6: M1*M2 is symmetric;
  M2*M1=(M1*M2)@ by A1,A5,A3,A4,MATRIX_3:22
    .=M1*M2 by A6;
  hence thesis;
end;
