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.
Proof k X step1.
End SUB_trans_proof.

Theorem SUB_trans : forall (A B C : Ens) (h : SUB A B) (k : SUB B C), SUB A C.