Section fct_CompleteSpace.
Context {T : Type} {U : CompleteSpace}.

(* Specifications: *)
Lemma filterlim_locally_cauchy :
forall {F} {FF : ProperFilter F} (f : T -> U),
(forall eps : posreal, exists P, F P /\ forall u v : T, P u -> P v -> ball (f u) eps (f v)) <->
exists y, filterlim f F (locally y).
Lemma filterlimi_locally_cauchy :
forall {F} {FF : ProperFilter F} (f : T -> U -> Prop),
F (fun x => (exists y, f x y) /\
(forall y1 y2, f x y1 -> f x y2 -> y1 = y2)) ->
((forall eps : posreal, exists P, F P /\
forall u v : T, P u -> P v -> forall u' v': U, f u u' -> f v v' -> ball u' eps v') <->
exists y, filterlimi f F (locally y)).