Definition choose (s : t) : option elt :=
    match choose s with
    | inleft (exist _ x _) => Some x
    | inright _ => None
    end.

(* Specifications: *)
Lemma choose_1 : forall (s : t) (x : elt), choose s = Some x -> In x s.
Lemma choose_2 : forall s : t, choose s = None -> Empty s.
Lemma choose_3 : forall s s' x x', choose s = Some x -> choose s' = Some x' -> Equal s s' -> E.eq x x'.