Section Integers_sect.
Inductive Integers : Ensemble nat :=
Integers_defn : forall x:nat, In nat Integers x.
Definition nat_po : PO nat.
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.

Lemma Finite_subset_has_lub :
forall X:Ensemble nat,
Finite nat X ->  exists m : nat, Upper_Bound nat nat_po X m.