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

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).