Proof total_th2_Out.
Section total_th3_proof.
Variable X Y Z F : Ens.
Hypothesis i : IN X Y.
Hypothesis j : IN Y Z.
Hypothesis k : IN Z F.
Let step1 : IN Y (UnionPlus F).
Proof UnionPlus_th2 Y Z F j k.
Let step2 : IN X (UnionPlus (UnionPlus F)).
Proof UnionPlus_th2 X Y (UnionPlus F) i step1.

Lemma total_th3_Out : IN X (Total F).
Proof UnionPlus_th1 X (UnionPlus (UnionPlus F)) step2.