reserve x, a, b, c for Real;

theorem
  for a, b, c, x being Complex holds a <> 0 & delta(a,b,c) = 0 &
  a * x^2 + b * x + c = 0 implies x = - b/(2 * a)
proof
  let a, b, c, x be Complex;
  assume that
A1: a <> 0 and
A2: delta(a,b,c) = 0 & a * x^2 + b * x + c = 0;
  (2 * a * x + b)^2 - 0 = 0 by A1,A2,Th14;
  then
A3: 2 * a * x + b = 0 by XCMPLX_1:6;
  2 * a <> 0 by A1;
  then x = (- b)/(2 * a) by A3,XCMPLX_1:89;
  hence thesis by XCMPLX_1:187;
end;
