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 O,N being Matrix of 3,REAL holds
  (N@ * O)@ * O * (N@ * O) = O@ * (N * O * N@) * O
  proof
    let O,N be Matrix of 3,REAL;
    (N@ * O)@ * O * (N@ * O) = (O@ * N@@) * O * (N@ * O) by MATRIXR2:30
                            .= (O@ * N) * O * (N@ * O) by MATRIXR2:29
                            .= (O@ * (N * O)) * (N@ * O) by MATRIXR2:27
                            .= (O@ * ((N * O) * (N@ * O))) by MATRIXR2:27
                            .= O@ * (((N * O) * N@) * O) by MATRIXR2:27;
    hence thesis by MATRIXR2:27;
  end;
