Section strictsub_trans1a.
Hypothesis m : EQ A C.
Let step2 : SUB C A. Proof EQ_backwards A C m.
Let step3 : SUB B A. Proof SUB_trans B C A k step2.
Let step4 : EQ A B. Proof Build_EQ A B h1 step3.
Proof h2 step4.
End strictsub_trans1a.
Let step5 : ~ EQ A C. Proof strictsub_trans1a_Out.
Proof Build_StrictSUB A C step1 step5.
End strictsub_trans1_proof.

Theorem StrictSUB_trans1 :
forall (A B C : Ens) (h : StrictSUB A B) (k : SUB B C), StrictSUB A C.