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
  for N being invertible Matrix of 3,F_Real
  for N1 being Matrix of 3,F_Real
  for M,NR being Matrix of 3,REAL st
  M = symmetric_3(1,1,-1,0,0,0) &
  N1 = M &
  NR = MXF2MXR(N~) holds
  (N@) * N1 * N = MXF2MXR((MXR2MXF(NR@))~) * M * MXF2MXR((MXR2MXF NR)~)
  proof
    let N be invertible Matrix of 3,F_Real;
    let N1 be Matrix of 3,F_Real;
    let M,NR be Matrix of 3,REAL;
    assume that
A1: M = symmetric_3(1,1,-1,0,0,0) and
A2: N1 = M and
A3: NR = MXF2MXR(N~);
    reconsider b = -1 as Element of F_Real by XREAL_0:def 1;
A4: b is non zero;
    reconsider a = 1 as Element of F_Real;
    a is non zero;
    then reconsider a = 1,b = -1 as non zero Element of F_Real by A4;
    reconsider N1 as invertible Matrix of 3,F_Real by A1,A2,Th44;
    reconsider M1 = (N@) * N1 * N as invertible Matrix of 3,F_Real;
    reconsider N2 = N1 as Matrix of 3,REAL;
A6: MXF2MXR((MXR2MXF NR)~) = (MXR2MXF NR)~ by MATRIXR1:def 2
                          .= (N~)~ by A3,ANPROJ_8:16
                          .= N by MATRIX_6:16;
A7: MXF2MXR((MXR2MXF(NR@))~) = N@
    proof
A8:   MXF2MXR((MXR2MXF(NR@))~) = (MXR2MXF(NR@))~ by MATRIXR1:def 2;
      MXR2MXF (NR@) = NR@ by MATRIXR1:def 1;
      then MXF2MXR((MXR2MXF(NR@))~) = ((N~)@)~ by A8,A3,MATRIXR1:def 2
                                   .= (N@)~~ by MATRIX_6:13;
      hence thesis by MATRIX_6:16;
    end;
    (N@) * N1 = MXF2MXR((MXR2MXF(NR@))~) * N2 by A7,ANPROJ_8:17;
    hence thesis by A2,A6,ANPROJ_8:17;
  end;
