reserve x, a, b, c for Real;

theorem
  a > 0 & delta(a,b,c) <= 0 implies a * x^2 + b * x + c >= 0
proof
  assume that
A1: a > 0 and
A2: delta(a,b,c) <= 0;
  - delta(a,b,c) >= -0 & 4 * a > 0 by A1,A2,XREAL_1:25,129;
  then (- delta(a,b,c))/(4 * a) >= 0 by XREAL_1:136;
  then - delta(a,b,c)/(4 * a) >= 0 by XCMPLX_1:187;
  then
A3: a * (x + b/(2 * a))^2 + - delta(a,b,c)/(4 * a) >= a * (x + b/(2 * a) )^2
  + 0 by XREAL_1:7;
  (x +b/(2 * a))^2 >= 0 by XREAL_1:63;
  then a * (x +b/(2 * a))^2 >= 0 by A1,XREAL_1:127;
  then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) >= 0 by A3,XXREAL_0:2;
  hence thesis by A1,Th1;
end;
