Section for_large.
Variable I : DirectedSet.
Definition eventually (P : DS_set I -> Prop) : Prop :=
exists i:DS_set I, forall j:DS_set I,
DS_ord i j -> P j.
Definition exists_arbitrarily_large (P: DS_set I -> Prop) :=
forall i:DS_set I, exists j:DS_set I,
DS_ord i j /\ P j.

Lemma not_eal_eventually_not: forall (P: DS_set I -> Prop),
~ exists_arbitrarily_large P ->
eventually (fun i:DS_set I => ~ P i).