(** * C is a CompleteSpace *)
Definition C_complete_lim (F : (C -> Prop) -> Prop) :=
(R_complete_lim (fun P => F (fun z => P (Re z))), R_complete_lim (fun P => F (fun z => P (Im z)))).

Lemma C_complete_ax1 :
forall F : (C -> Prop) -> Prop,
ProperFilter F ->
(forall eps : posreal, exists x : C, F (ball x eps)) ->
forall eps : posreal, F (ball (C_complete_lim F) eps).