Section Locally_fct.
Context {T : Type} {U : UniformSpace}.

(* Specifications: *)
Lemma filterlim_locally :
forall {F} {FF : Filter F} (f : T -> U) y,
filterlim f F (locally y) <->
forall eps : posreal, F (fun x => ball y eps (f x)).
Lemma filterlimi_locally :
forall {F} {FF : Filter F} (f : T -> U -> Prop) y,
filterlimi f F (locally y) <->
forall eps : posreal, F (fun x => exists z, f x z /\ ball y eps z).
Lemma filterlim_locally_close :
forall {F} {FF: ProperFilter F} (f : T -> U) l l',
filterlim f F (locally l) ->  filterlim f F (locally l') ->
close l l'.
Lemma filterlimi_locally_close :
forall {F} {FF: ProperFilter F} (f : T -> U -> Prop) l l',
F (fun x => forall y1 y2, f x y1 -> f x y2 -> y1 = y2) ->
filterlimi f F (locally l) ->  filterlimi f F (locally l') ->
close l l'.