Inductive neighborhood_net_DS_set : Type :=
  | intro_neighborhood_net_DS :
    forall (U:Ensemble (point_set X)) (y:point_set X),
    open U -> In U x -> In U y -> neighborhood_net_DS_set.Inductive neighborhood_net_DS_set : Type :=
  | intro_neighborhood_net_DS :
    forall (U:Ensemble (point_set X)) (y:point_set X),
    open U -> In U x -> In U y -> neighborhood_net_DS_set.Definition neighborhood_net_DS_ord
  (Uy Vz:neighborhood_net_DS_set) : Prop :=
  match Uy, Vz with
  | intro_neighborhood_net_DS U _ _ _ _,
    intro_neighborhood_net_DS V _ _ _ _ =>
    Included V U
  end.Definition neighborhood_net_DS : DirectedSet.Definition neighborhood_net : Net neighborhood_net_DS X :=
  fun (x:neighborhood_net_DS_set) => match x with
  | intro_neighborhood_net_DS _ y _ _ _ => y
  end.

Lemma neighborhood_net_limit: net_limit neighborhood_net x.