Section fct_CompleteSpace.
Context {T : Type} {U : CompleteSpace}.
Definition lim_fct (F : ((T -> U) -> Prop) -> Prop) (t : T) :=
lim (fun P => F (fun g => P (g t))).

Lemma complete_cauchy_fct :
forall (F : ((T -> U) -> Prop) -> Prop),
ProperFilter F ->
(forall eps : posreal, exists f : T -> U, F (ball f eps)) ->
forall eps : posreal, F (ball (lim_fct F) eps).