theorem Th54:
  for ra being Real for M,N being Matrix of 3,F_Real
  for MR,NR being Matrix of 3,REAL st MR = M & NR = N &
  N is symmetric & MR = ra * NR holds M is symmetric
  proof
    let ra be Real;
    let M,N be Matrix of 3,F_Real;
    let MR,NR be Matrix of 3,REAL;
    assume that
A1: MR = M & NR = N and
A2: N is symmetric and
A3: MR = ra * NR;
    width NR > 0 by MATRIX_0:23;
    then M@ = ra * NR@ by A1,A3,MATRIXR1:30
           .= M by A1,A2,A3,MATRIX_6:def 5;
    hence thesis by MATRIX_6:def 5;
  end;
