Section strictsub_next1.
Hypothesis k : EQ A N.
Let step2 : SUB N A := EQ_backwards A N k.
Let step3 : IN A N := anything_in_next_of_itself A.
Let step4 : IN A A := step2 A step3.
Definition strictsub_next1_Out : False := h step4.
End strictsub_next1.
Let step5 : ~ EQ A N := strictsub_next1_Out.
Definition strictsub_next_Out : StrictSUB A N :=
Build_StrictSUB A N step1 step5.
End strictsub_next_proof.

Theorem some_things_strictsub_their_nexts :
forall (A : Ens) (h : not_in_itself A), StrictSUB A (Next A).