reserve x, a, b, c for Real;

theorem Th17:
  a < 0 & delta(a,b,c) > 0 implies (- b + sqrt delta(a,b,c))/(2 *
  a) < (- b - sqrt delta(a,b,c))/(2 * a)
proof
  assume that
A1: a < 0 and
A2: delta(a,b,c) > 0;
  sqrt delta(a,b,c) > 0 by A2,SQUARE_1:25;
  then 2 * sqrt delta(a,b,c) > 0 by XREAL_1:129;
  then sqrt delta(a,b,c) + sqrt delta(a,b,c) > 0;
  then sqrt delta(a,b,c) > - sqrt delta(a,b,c) by XREAL_1:59;
  then
A3: - b + sqrt delta(a,b,c) > - b + - sqrt delta(a,b,c) by XREAL_1:6;
  2 * a < 0 by A1,XREAL_1:132;
  hence thesis by A3,XREAL_1:75;
end;
