Section total_th2_proof.
Variable X Y F : Ens.
Hypothesis i : IN X Y.
Hypothesis j : IN Y F.
Let step1 : IN X (UnionPlus F).
Proof UnionPlus_th2 X Y F i j.
Let step2 : IN X (UnionPlus (UnionPlus F)).
Proof UnionPlus_th1 X (UnionPlus F) step1.

Lemma total_th2_Out : IN X (Total F).
Proof UnionPlus_th1 X (UnionPlus (UnionPlus F)) step2.