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 Th53:
  for ra being non zero Real
  for N,O,M being Matrix of 3,3,REAL st N is invertible &
  M = ra * O & M = N@ * O * N holds Inv(N)@ * O * Inv(N) = 1/ra * O
  proof
    let ra be non zero Real;
    let N,O,M be Matrix of 3,3,REAL;
    assume that
A1: N is invertible and
A3: M = ra * O and
A4: M = N@ * O * N;
    reconsider NI = Inv(N), NIT = (Inv(N))@ as Matrix of 3,3,REAL;
A5: (Inv(N))@ = Inv(N@) & N@ is invertible by A1,Th52;
A6: len NI = 3 & width NI = 3 & width (O * N) = 3 & len (O * N) = 3
       by MATRIX_0:24;
    reconsider ira = 1/ra as Real;
    reconsider NTON = N@ * O * N as Matrix of REAL;
A7: len NTON = 3 by MATRIX_0:24;
A8: width(NI@) = 3 by MATRIX_0:24;
    O = 1 * O by MATRIXR1:32
     .= (1/ra * ra) * O by XCMPLX_1:87
     .= ira * (N@ * O * N) by A3,A4,MATRIXR2:11;
    then NI@ * O * NI = ira * (NI@ * (N@ * O * N)) * NI by A8,A7,MATRIXR1:40
                     .= ira * (NIT * (N@ * (O * N))) * NI by MATRIXR2:27
                     .= ira * ((NIT * N@) * (O * N)) * NI by MATRIXR2:27
                     .= ira * (1_Rmatrix(3) * (O * N)) * NI
                        by A5,MATRIXR2:def 6
                     .= ira * (O * N) * NI by MATRIXR2:70
                     .= ira * (O * N * NI) by A6,MATRIXR1:41
                     .= ira * (O * (N * NI)) by MATRIXR2:27
                     .= ira * (O * 1_Rmatrix(3)) by A1,MATRIXR2:def 6
                     .= ira * O by MATRIXR2:71;
    hence thesis;
  end;
