reserve x, a, b, c for Real;

theorem
  ( for x holds a * x^2 + b * x + c >= 0 ) & a > 0 implies delta(a,b,c) <= 0
proof
  assume that
A1: for x holds a * x^2 + b * x + c >= 0 and
A2: a > 0;
  a * (- b/(2 * a))^2 + b * (- b/(2 * a)) + c >= 0 by A1;
  then (2 * a * (- b/(2 * a)) + b)^2 - delta(a,b,c) >= 0 by A2,Th6;
  then
A3: (- (2 * a) * (b/(2 * a)) + b)^2 - delta(a,b,c) >= 0;
  2 * a <> 0 by A2;
  then (- b + b)^2 - delta(a,b,c) >= 0 by A3,XCMPLX_1:87;
  then - delta(a,b,c) >= -0;
  hence thesis by XREAL_1:24;
end;
