(******** transitivity **********)
Section SUB_trans_proof.
Variable A : Ens.
Variable B : Ens.
Variable C : Ens.
Hypothesis h : SUB A B.
Hypothesis k : SUB B C.
Variable X : Ens.
Hypothesis m : IN X A.
Let step1 : IN X B.
Proof h X m.

Lemma SUB_trans_proof_Out : IN X C.
Proof k X step1.