Proof
fun (X U F : Ens) (i : IN X U) (j : IN U F) =>
union_th2 F (Union F) X (Union_th1 F X U i j).
Section total_th1_proof.
Variable X F : Ens.
Hypothesis i : IN X F.
Let step1 : IN X (UnionPlus F).
Proof UnionPlus_th1 X F i.
Let step2 : IN X (UnionPlus (UnionPlus F)).
Proof UnionPlus_th1 X (UnionPlus F) step1.

(* Specifications: *)
Lemma total_th1_Out : IN X (Total F).
Proof UnionPlus_th1 X (UnionPlus (UnionPlus F)) step2.