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

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).