Definition R_AbsRing_mixin :=
AbsRing.Mixin _ _ Rabs_R0 Rabs_m1 Rabs_triang (fun x y => Req_le _ _ (Rabs_mult x y)) Rabs_eq_0.
Canonical R_AbsRing :=
AbsRing.Pack R (AbsRing.Class _ _ R_AbsRing_mixin) R.
Definition R_UniformSpace_mixin :=
AbsRing_UniformSpace_mixin R_AbsRing.
Canonical R_UniformSpace :=
UniformSpace.Pack R R_UniformSpace_mixin R.
Definition R_complete_lim (F : (R -> Prop) -> Prop) : R :=
Lub_Rbar (fun x : R => F (ball (x + 1) (mkposreal _ Rlt_0_1))).

Lemma R_complete_ax1 :
forall F : (R -> Prop) -> Prop,
ProperFilter' F ->
(forall eps : posreal, exists x : R, F (ball x eps)) ->
forall eps : posreal, F (ball (R_complete_lim F) eps).