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 Th55:
  for ra being Real for O,M being Matrix of 3,REAL
  st O = symmetric_3(1,1,-1,0,0,0) & M = ra * O holds
  O * M = ra * 1_Rmatrix(3) & M * O = ra * 1_Rmatrix(3)
  proof
    let ra be Real;
    let O,M be Matrix of 3,REAL;
    assume that
A1: O = symmetric_3(1,1,-1,0,0,0) and
A2: M = ra * O;
    1_Rmatrix(3) = MXF2MXR 1.(F_Real,3) by MATRIXR2:def 2; then
A3: MXR2MXF 1_Rmatrix(3) = 1.(F_Real,3) by ANPROJ_8:16;
A4: O * O = symmetric_3(1,1,-1,0,0,0) * symmetric_3(1,1,-1,0,0,0)
              by A1,ANPROJ_8:17
         .= 1_Rmatrix(3) by Th43,A3,MATRIXR1:def 1;
    len O = 3 & width O = 3 by MATRIX_0:23;
    hence thesis by A4,A2,MATRIXR1:40,41;
  end;
