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 + 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;
  thus 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 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_1:82;
    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_1:134;
    hence thesis by XREAL_1:47,48;
  end;
  assume x < (- b + sqrt delta(a,b,c))/(2 * a) or x > (- b - sqrt delta(a,b,
  c))/(2 * a);
  then
A3: x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 or x - (- b - sqrt delta(a,b,
  c))/(2 * a) > 0 by XREAL_1:49,50;
  (- b + sqrt delta(a,b,c))/(2 * a) < (- b - sqrt delta(a,b,c))/(2 * a)
  by A1,A2,Th17;
  then
  x - (- b + sqrt delta(a,b,c))/(2 * a) > x - (- b - sqrt delta(a,b,c))/(
  2 * a) by XREAL_1:10;
  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 A3,XXREAL_0:2;
  then
  (x - (- b + sqrt delta(a,b,c))/(2 * a)) * (x - (- b - sqrt delta(a,b,c)
  )/(2 * a)) > 0 by XREAL_1:129,130;
  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;
