Definition complete : Prop :=
  forall x:nat->X, cauchy x ->
    exists x0:X, net_limit x x0 (I:=nat_DS)
      (X:=MetricTopology d d_metric).

(* Specifications: *)
Lemma d_restriction_metric: metric d_restriction.
Lemma closed_subset_of_complete_is_complete: complete d d_metric -> closed F (X:=MetricTopology d d_metric) -> complete d_restriction d_restriction_metric.
Lemma complete_subset_is_closed: complete d_restriction d_restriction_metric -> closed F (X:=MetricTopology d d_metric).