Proof
fun (F X : Ens) (i : IN X (Union F)) =>
nested_IN_bc X (Union_intermediate F X i) F (Union_intermediate_pr F X i).
Section Union_th1_proof.
Variable F : Ens.
Variable X : Ens.
Variable U : Ens.
Hypothesis h : IN X U.
Hypothesis k : IN U F.
Let step1 : nested_IN X U F.
Proof Build_nested_IN X U F h k.
Let step2 : EXISTS (fun U1 : Ens => nested_IN X U1 F).
Proof EXISTS_th1 (fun U1 : Ens => nested_IN X U1 F) U step1.

(* Specifications: *)
Lemma union_th1_OutA : IN X (Union F).
Proof Union_pr1 F X step2.