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 le_total_order : Totally_ordered nat nat_po Integers.