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.

Lemma eventually_and: forall (P Q: DS_set I -> Prop),
eventually P -> eventually Q ->
eventually (fun i:DS_set I => P i /\ Q i).