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

(* Specifications: *)
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Lemma continuity_P2 :
forall (f:R -> R) (D:R -> Prop),
continuity f -> open_set D -> open_set (image_rec f D).
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Lemma continuity_P3 :
forall f:R -> R,
continuity f <->
(forall D:R -> Prop, open_set D -> open_set (image_rec f D)).
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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)).