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

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