Definition partial_WO_chain_ub: forall C:Ensemble partial_WO,
  chain partial_WO_ord C -> partial_WO.

Lemma partial_WO_chain_ub_correct: forall (C:Ensemble partial_WO) (c:chain partial_WO_ord C), forall WO:partial_WO, In C WO -> partial_WO_ord WO (partial_WO_chain_ub C c).