reserve x, a, b, c for Real;

theorem
  a < 0 & delta(a,b,c) = 0 implies ( a * x^2 + b * x + c < 0 iff x <> -
  b/(2 * a) )
proof
  assume that
A1: a < 0 and
A2: delta(a,b,c) = 0;
A3: 2 * a <> 0 by A1;
  thus a * x^2 + b * x + c < 0 implies x <> - b/(2 * a)
  proof
    assume a * x^2 + b * x + c < 0;
    then (2 * a * x + b)^2 - 0 > 0 by A1,A2,Th9;
    then 2 * a * x <> - b;
    then x <> (- b)/(2 * a) by A3,XCMPLX_1:87;
    hence thesis by XCMPLX_1:187;
  end;
  assume x <> - b/(2 * a);
  then x <> (- b)/(2 * a) by XCMPLX_1:187;
  then 2 * a * x + b <> 0 by A3,XCMPLX_1:89;
  then (2 * a * x + b)^2 - delta(a,b,c) > 0 by A2,SQUARE_1:12;
  hence thesis by A1,Th23;
end;
