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 Th13:
  a <> 0 & -1 < a < 1 & b = (2 + sqrt delta(a * a,-2,1)) / (2 * a * a)
  implies (1 + a * a) * b * b - 2 * b + 1 - b * b = 0
  proof
    assume that
A1: a <> 0 and
A2: -1 < a < 1 and
A3: b = (2 + sqrt delta(a * a,-2,1)) / (2 * a * a);
    set x1 = (- (-2) - sqrt(delta(a * a,-2,1)))/(2 * (a * a)),
        x2 = (- (-2) + sqrt(delta(a * a,-2,1)))/(2 * (a * a));
A4: 0 <= 1 - a^2
    proof
      a^2 * 1^2 <= 1 by A2,SQUARE_1:53;
      then a^2 < 1 by A2,SQUARE_1:41,XXREAL_0:1;
      then a^2 - a^2 < 1 - a^2 by XREAL_1:9;
      hence thesis;
    end;
    delta(a * a,-2,1) >= 0
    proof
      delta(a * a,-2,1) = (-2)^2 - 4 * (a * a) * 1 by QUIN_1:def 1
      .= 4 * (1 - a^2);
      hence thesis by A4;
    end; then
    (a * a) * b^2 + (- 2) * b + 1 = (a * a) * (b - x1) * (x2 - x2)
      by A3,A1,QUIN_1:16
                                 .= 0;
    hence thesis;
  end;
