(* Specifications: *)
Lemma inverses_of_nats_approach_0: forall eps:R, eps > 0 -> exists n:nat, (n > 0)%nat /\ / (INR n) < eps.
Lemma Z_interpolation: forall x y:R, y > x+1 -> exists n:Z, x < IZR n < y.
Lemma interior_open : open interior.
Lemma interior_deflationary : Included interior S.
Lemma interior_fixes_open: open S -> interior = S.
Lemma interior_maximal: forall U:Ensemble (point_set X), open U -> Included U S -> Included U interior.
Lemma closure_closed : closed closure.
Lemma closure_inflationary : Included S closure.
Lemma closure_fixes_closed : closed S -> closure = S.
Lemma closure_minimal: forall F:Ensemble (point_set X), closed F -> Included S F -> Included closure F.
Lemma interior_increasing: forall S T:Ensemble (point_set X), Included S T -> Included (interior S) (interior T).
Lemma interior_intersection: forall S T:Ensemble (point_set X), interior (Intersection S T) = Intersection (interior S) (interior T).
Lemma interior_union: forall S T:Ensemble (point_set X), Included (Union (interior S) (interior T)) (interior (Union S T)).
Lemma complement_inclusion: forall {Y:Type} (S T:Ensemble Y), Included S T -> Included (Complement T) (Complement S).
Lemma interior_complement: forall S:Ensemble (point_set X), interior (Complement S) = Complement (closure S).
Lemma closure_increasing: forall S T:Ensemble (point_set X), Included S T -> Included (closure S) (closure T).
Lemma closure_complement: forall S:Ensemble (point_set X), closure (Complement S) = Complement (interior S).
Lemma closure_union: forall S T:Ensemble (point_set X), closure (Union S T) = Union (closure S) (closure T).
Lemma closure_intersection: forall S T:Ensemble (point_set X), Included (closure (Intersection S T)) (Intersection (closure S) (closure T)).
Lemma meets_every_open_neighborhood_impl_closure: forall (S:Ensemble (point_set X)) (x:point_set X), (forall U:Ensemble (point_set X), open U -> In U x -> Inhabited (Intersection S U)) -> In (closure S) x.
Lemma closure_impl_meets_every_open_neighborhood: forall (S:Ensemble (point_set X)) (x:point_set X), In (closure S) x -> forall U:Ensemble (point_set X), open U -> In U x -> Inhabited (Intersection S U).
Lemma completion_exists: forall (X:Type) (d:X->X->R) (d_metric:metric d), exists Y:Type, exists i:X->Y, exists d':Y->Y->R, exists d'_metric:metric d', injective i /\ dense (Im Full_set i) (X:=MetricTopology d' d'_metric) /\ (forall x1 x2:X, d' (i x1) (i x2) = d x1 x2) /\ complete d' d'_metric.
Lemma subspace_inc_takes_closed_to_closed: forall G:Ensemble (point_set (SubspaceTopology F)), closed G -> closed (Im G (subspace_inc F)).
Lemma Rle_order: order Rle.
Lemma Rplus_continuous: continuous_2arg Rplus (X:=RTop) (Y:=RTop) (Z:=RTop).