Infix "=_D" := eq_Dom (at level 70, no associativity).

(* Specifications: *)
Lemma open_set_P1 : forall D:R -> Prop, open_set D <-> D =_D interior D.
Lemma closed_set_P1 : forall D:R -> Prop, closed_set D <-> D =_D adherence D.
Lemma neighbourhood_P1 :
forall (D1 D2:R -> Prop) (x:R),
included D1 D2 -> neighbourhood D1 x -> neighbourhood D2 x.
Lemma open_set_P2 :
forall D1 D2:R -> Prop,
open_set D1 -> open_set D2 -> open_set (union_domain D1 D2).
Lemma open_set_P3 :
forall D1 D2:R -> Prop,
open_set D1 -> open_set D2 -> open_set (intersection_domain D1 D2).
Lemma open_set_P4 : open_set (fun x:R => False).
Lemma open_set_P5 : open_set (fun x:R => True).
Lemma disc_P1 : forall (x:R) (del:posreal), open_set (disc x del).
Lemma continuity_P1 :
forall (f:R -> R) (x:R),
continuity_pt f x <->
(forall W:R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
neighbourhood V x /\ (forall y:R, V y -> W (f y))).