Section CompleteSpace1.
Context {T : CompleteSpace}.
Definition lim : ((T -> Prop) -> Prop) -> T := CompleteSpace.lim _ (CompleteSpace.class T).
Definition iota (P : T -> Prop) := lim (fun A => (forall x, P x -> A x)).
End CompleteSpace1.

Lemma cauchy_distance :
forall {T : UniformSpace} {F} {FF : ProperFilter F},
(forall eps : posreal, exists x, F (ball x eps)) <->
(forall eps : posreal, exists P, F P /\ forall u v : T, P u -> P v -> ball u eps v).