(** * 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).