Definition nat_po : PO nat.

(* 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.