apply Definition_of_PO with (Carrier_of := Integers) (Rel_of := le);
auto with sets arith.
- apply Inhabited_intro with (x := 0).
apply Integers_defn.
- exact le_Order.
Defined.

(* Specifications: *)
Lemma le_total_order : Totally_ordered nat nat_po Integers.
Lemma Finite_subset_has_lub :
forall X:Ensemble nat,
Finite nat X ->  exists m : nat, Upper_Bound nat nat_po X m.
Lemma Integers_has_no_ub :
~ (exists m : nat, Upper_Bound nat nat_po Integers m).
Lemma Integers_infinite : ~ Finite nat Integers.
End Integers_sect.