Definition Add x s s' := forall y, In y s' <-> O.eq x y \/ In y s.

Lemma lt_add_lt : forall x1 x2 s1 s1' s2 s2', Above x1 s1 -> Above x2 s2 -> Add x1 s1 s1' -> Add x2 s2 s2' -> O.lt x1 x2 -> lt s1' s2'.