reserve x, a, b, c for Real;

theorem
  a > 0 & delta(a,b,c) > 0 implies ( a * x^2 + b * x + c < 0 iff (- b -
  sqrt delta(a,b,c))/(2 * a) < x & x < (- b + sqrt delta(a,b,c))/(2 * a) )
proof
  assume that
A1: a > 0 and
A2: delta(a,b,c) > 0;
  thus a * x^2 + b * x + c < 0 implies (- b - sqrt delta(a,b,c))/(2 * a) < x &
  x < (- b + sqrt delta(a,b,c))/(2 * a)
  proof
    assume a * x^2 + b * x + c < 0;
    then
    a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(a
    ,b,c))/(2 * a)) < 0 by A1,A2,Th16;
    then
    a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(
    a,b,c))/(2 * a))) < 0;
    then
    (x - (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(a,b,c
    ))/(2 * a)) < 0/a by A1,XREAL_1:81;
    then
    x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 & x - (- b + sqrt delta(a,b,
c))/(2 * a) < 0 or x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 & x - (- b + sqrt
    delta(a,b,c))/(2 * a) > 0 by XREAL_1:133;
    then
    x > (- b - sqrt delta(a,b,c))/(2 * a) & x < (- b + sqrt delta(a,b,c))/
(2 * a) or (- b + sqrt delta(a,b,c))/(2 * a) > (- b - sqrt delta(a,b,c))/(2 * a
) & x < (- b - sqrt delta(a,b,c))/(2 * a) & x > (- b + sqrt delta(a,b,c))/(2 *
    a) by A1,A2,Th25,XREAL_1:47,48;
    hence thesis by XXREAL_0:2;
  end;
  assume (- b - sqrt delta(a,b,c))/(2 * a) < x & x < (- b + sqrt delta(a,b,c
  ))/(2 * a );
  then
  x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 & x - (- b + sqrt delta(a,b,c
  ))/(2 * a) < 0 by XREAL_1:49,50;
  then
  (x - (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(a,b,c)
  )/(2 * a)) < 0 by XREAL_1:132;
  then
  a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(a
  ,b,c))/(2 * a))) < 0 by A1,XREAL_1:132;
  then
  a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) * (x - (- b + sqrt delta(a,
  b,c))/(2 * a)) < 0;
  hence thesis by A1,A2,Th16;
end;
