Section EQ_trans_proof.
Variable A : Ens.
Variable B : Ens.
Variable C : Ens.
Hypothesis h : EQ A B.
Hypothesis k : EQ B C.
Let step1 : SUB A B.
Proof EQ_forwards A B h.
Let step2 : SUB B A.
Proof EQ_backwards A B h.
Let step3 : SUB B C.
Proof EQ_forwards B C k.
Let step4 : SUB C B.
Proof EQ_backwards B C k.
Let step5 : SUB A C.
Proof SUB_trans A B C step1 step3.
Let step6 : SUB C A.
Proof SUB_trans C B A step4 step2.
Proof Build_EQ A C step5 step6.
End EQ_trans_proof.

Theorem EQ_trans : forall (A B C : Ens) (h : EQ A B) (k : EQ B C), EQ A C.