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 * b)^2 + b^2 = 1 implies b = 1 / sqrt(1 + a^2) or
     b = (-1) / sqrt(1 + a^2)
  proof
    assume a1: (a * b)^2 + b^2 = 1;
    b^2 = (a^2 + 1) * b^2 / (a^2 + 1) by XCMPLX_1:89,Lem03
       .= 1 / (a^2 + 1) by a1;
    hence thesis by Lem04;
  end;
