Proof total_th3_Out.
Section total_th4_proof.
Variable X Y Z W F : Ens.
Hypothesis i : IN X Y.
Hypothesis j : IN Y Z.
Hypothesis k : IN Z W.
Hypothesis l : IN W F.
Let step1 : IN Z (UnionPlus F).
Proof UnionPlus_th2 Z W F k l.
Let step2 : IN Y (UnionPlus (UnionPlus F)).
Proof UnionPlus_th2 Y Z (UnionPlus F) j step1.

Lemma total_th4_Out : IN X (Total F).
Proof UnionPlus_th2 X Y (UnionPlus (UnionPlus F)) i step2.