(** * Emptiness is decidable *)
Definition Empty (E : R -> Prop) :=
Lub_Rbar (fun x => x = 0 \/ E x) = Glb_Rbar (fun x => x = 0 \/ E x)
/\ Lub_Rbar (fun x => x = 1 \/ E x) = Glb_Rbar (fun x => x = 1 \/ E x).

Lemma uniqueness_dec P : (exists ! x : R, P x) -> {x : R | P x}.