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)).

Lemma iota_correct_weak :
forall P : T -> Prop,
(forall x y, P x -> P y -> close x y) ->
forall x, P x -> close (iota P) x.