Definition extend_strictly_partial_WO: forall (WO:partial_WO)
  (a:T), ~ In (pwo_S WO) a -> partial_WO.

(* Specifications: *)
Lemma extend_strictly_partial_WO_correct: forall (WO:partial_WO) (x:T) (ni:~ In (pwo_S WO) x), partial_WO_ord WO (extend_strictly_partial_WO WO x ni).
Lemma premaximal_partial_WO_is_full: forall WO:partial_WO, premaximal partial_WO_ord WO -> pwo_S WO = Full_set.
Theorem well_orderable: exists R:relation T, well_order R.
Lemma WO_implies_AC: forall (A B:Type) (R: A -> B -> Prop) (WO:relation B), well_order WO -> (forall x:A, exists y:B, R x y) -> exists f:A->B, forall x:A, R x (f x).