simple induction E; intros A f HR.
simpl in |- *; split.
intros P; exists P; split.
intros c; elim c; intros a p.
exists (dep_i A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
intros c; elim c; intros a p.
exists (dep_i'' A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
intros P; exists P; split.
intros c; elim c; intros a p.
exists (dep_i A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
intros c; elim c; intros a p.
exists (dep_i'' A (fun a0 : A => P a0) a p); simpl in |- *; auto with zfc.
Qed.
(* The set of small sets                        *)
Definition Big := sup Ens' inj.

Theorem Big_is_big : forall E : Ens', IN (inj E) Big.
intros E; unfold Big in |- *; simpl in |- *; exists E; auto with zfc.