(*****  StrictSUB *****)
Record StrictSUB (A B : Ens) : Prop :=
{StrictSUB_sub : SUB A B; StrictSUB_neq : ~ EQ A B}.
Section strictsub_trans1_proof.
Variable A : Ens.
Variable B : Ens.
Variable C : Ens.
Hypothesis h : StrictSUB A B.
Hypothesis k : SUB B C.
Let h1 : SUB A B.
Proof StrictSUB_sub A B h.
Let h2 : ~ EQ A B.
Proof StrictSUB_neq A B h.
Let step1 : SUB A C.
Proof SUB_trans A B C h1 k.
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.

Lemma strictsub_trans1a_Out : False.
Proof h2 step4.