reserve x, a, b, c for Real;

theorem
  a <> 0 & delta(a,b,c) >= 0 & a * x^2 + b * x + c = 0 implies x = (- b
  - sqrt delta(a,b,c))/(2 * a) or x = (- b + sqrt delta(a,b,c))/(2 * a)
proof
  assume that
A1: a <> 0 and
A2: delta(a,b,c) >= 0 and
A3: a * x^2 + b * x + c = 0;
  (2 * a * x + b)^2 - delta(a,b,c) = 0 by A1,A3,Th14;
  then (2 * a * x + b)^2 = (sqrt delta(a,b,c))^2 by A2,SQUARE_1:def 2;
  then
A4: 2 * a * x + b = sqrt delta(a,b,c) or 2 * a * x + b = - sqrt delta(a, b,c
  ) by Lm1;
  2 * a <> 0 by A1;
  hence thesis by A4,XCMPLX_1:89;
end;
