reserve x, a, b, c for Real;

theorem Th16:
  a <> 0 & delta(a,b,c) >= 0 implies a * x^2 + b * x + c = a * (x
- (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(a,b,c))/(2 * a))
proof
  assume that
A1: a <> 0 and
A2: delta(a,b,c) >= 0;
  a * x^2 + b * x + c = a * (x + b/(2 * a))^2 - 1 * (delta(a,b,c)/(4 * a))
  by A1,Th1
    .= a * (x + b/(2 * a))^2 - (a * (1/a)) * (delta(a,b,c)/(4 * a)) by A1,
XCMPLX_1:106
    .= a * ((x + b/(2 * a))^2 - (1/a) * (delta(a,b,c)/(4 * a)))
    .= a * ((x + b/(2 * a))^2 - (delta(a,b,c) * 1)/((4 * a) * a)) by
XCMPLX_1:76
    .= a * ((x + b/(2 * a))^2 - (sqrt delta(a,b,c))^2/(2 * a)^2) by A2,
SQUARE_1:def 2
    .= a * ((x + b/(2 * a))^2 - (sqrt delta(a,b,c)/(2 * a))^2) by XCMPLX_1:76
    .= a * (x - (- b/(2 * a) + sqrt delta(a,b,c)/(2 * a))) * (x - (- b/(2 *
  a) - sqrt delta(a,b,c)/(2 * a)))
    .= a * (x - ((- b)/(2 * a) + sqrt delta(a,b,c)/(2 * a))) * (x - (- b/(2
  * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:187
    .= a * (x - ((- b)/(2 * a) + sqrt delta(a,b,c)/(2 * a))) * (x - ((- b)/(
  2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:187
    .= a * (x - (- b + sqrt delta(a,b,c))/(2 * a)) * (x - ((- b)/(2 * a) -
  sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:62
    .= a * (x - (- b + sqrt delta(a,b,c))/(2 * a)) * (x - (- b - sqrt delta(
  a,b,c))/(2 * a)) by XCMPLX_1:120;
  hence thesis;
end;
