reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

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;
