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 & -1 < c < 1 implies ex d,e,f st
  e = d * c * a + (1 - d) * (-b) &
  f = d * c * b + (1 - d) * a & e^2 + f^2 = d^2
  proof
    assume that
A1: a^2 + b^2 = 1 and
A2: -1 < c < 1;
    consider d be Real such that
A3: (1 + c * c) * d * d - 2 * d + 1 - d * d = 0 by A2,Lem07;
    reconsider e = d * c * a + (1 - d) * (-b),
               f = d * c * b + (1 - d) * a as Real;
    e^2 + f^2 = (d * c)^2 * (a^2 + b^2) + (((1 - d) * b)^2 + ((1 - d) *a)^2)
             .= (d * c)^2 * 1 + (1 - d)^2 * 1 by A1,Lem06
             .= d^2 by A3;
    hence thesis;
  end;
