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;

theorem
  a <> 0 & b^2 = 1 + a * a implies
  a * (1/b) * a * ((-1)/b) + (1/b) * ((-1)/b) = -1
  proof
    assume that
A1: a <> 0 and
A2: b^2 = 1 + a * a;
A3: b <> 0
    proof
      assume b = 0;
      then a^2 = -1 by A2;
      hence contradiction by A1,SQUARE_1:12;
    end;
    a * (1/b) * a * ((-1)/b) + (1/b) * ((-1)/b) = (1/b) * (((-1) * 1)/b)
                                                   * (a * a + 1)
                                               .= (1/b) * ((-1) * (1/b))
                                                   * (a * a + 1) by XCMPLX_1:74
                                               .= (-1) * ((1/b)^2 * b^2) by A2
                                               .= (-1) * 1 by A3,Lem05;
    hence thesis;
  end;
