Definition image_rec (f:R -> R) (D:R -> Prop) (x:R) : Prop := D (f x).

(* Specifications: *)
Lemma continuity_P2 : forall (f:R -> R) (D:R -> Prop), continuity f -> open_set D -> open_set (image_rec f D).
Lemma continuity_P3 : forall f:R -> R, continuity f <-> (forall D:R -> Prop, open_set D -> open_set (image_rec f D)).
Theorem Rsepare : forall x y:R, x <> y -> exists V : R -> Prop, (exists W : R -> Prop, neighbourhood V x /\ neighbourhood W y /\ ~ (exists y : R, intersection_domain V W y)).