Definition Subset (s s': t) := Subset' (proj1_sig s) (proj1_sig s').
Definition Empty (s: t) := Empty' (proj1_sig s).
Definition For_all (P : elt -> Prop) (s: t) := For_all' P (proj1_sig s).
Definition Exists (P : elt -> Prop) (s: t) := Exists' P  (proj1_sig s).
Definition eq : t -> t -> Prop := Equal.

Lemma fold_spec : forall s (A : Type) (i : A) (f : elt -> A -> A),
fold _ f s i = List.fold_left (fun a e => f e a) (elements s) i.