reserve x, a, b, c for Real;

theorem Th14:
  for a, b, c, x being Complex holds a <> 0 & a * x^2 + b *
  x + c = 0 implies (2 * a * x + b)^2 - delta(a,b,c) = 0
proof
  let a, b, c, x be Complex;
  assume that
A1: a <> 0 and
A2: a * x^2 + b * x + c = 0;
A3: 4 * a <> 0 by A1;
  a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) = 0 by A1,A2,Th1;
  then
A4: ((2 * a) * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a
  )) = 0;
  2 * a <> 0 by A1;
  then (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) = 0 by A4,
XCMPLX_1:87;
  hence thesis by A3,XCMPLX_1:87;
end;
