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^2 + b^2 = 1 & (c * a)^2 + (c * b)^2 = 1 implies c = 1 or c = -1
  proof
    assume that
A1: a^2 + b^2 = 1 and
A2: (c * a)^2 + (c * b)^2 = 1;
    1 = (c * c)  *(a^2 + b * b) by A2
     .= c^2 by A1;
    hence thesis by SQUARE_1:41;
  end;
