Definition NG_existence_prop (F : R -> Prop) (P : R -> R -> Prop) :=
  forall x d u, ~F x -> Rnd_DN_pt F x d -> Rnd_UP_pt F x u -> P x u \/ P x d.

Theorem satisfies_any_imp_NA : forall F : R -> Prop, satisfies_any F -> round_pred (Rnd_NA_pt F).