Definition Qabs (q:Q):Q:= 
  match q with 
  | Zero => Zero
  | Qpos qp => Qpos qp
  | Qneg qp => Qpos qp
  end.

Lemma lower_bound_affine_base_interval_twice:forall a b c d x, -Qone<=x -> x<=Qone -> -Qabs (a*c) + -Qabs (a*d+c*b)+b*d <= (a*x+b)*(c*x+d).