Proof strictsub_next_Out.
(* now we prove by induction the in_then_strictsub property for Naturals *)
Section in_then_strictsub_Zero_proof.
Variable X : Ens.
Hypothesis h : IN X Zero.
Let step1 : False := EmptySet_th1 X h.
Definition in_iff_strictsub_Zero_Out : StrictSUB X Zero :=
false_implies_everything (StrictSUB X Zero) step1.
End in_then_strictsub_Zero_proof.

Theorem in_then_strictsub_Zero :
forall X : Ens, IN X Zero -> StrictSUB X Zero.