induction ls; intuition; simpl in *.
destruct i.
omega.
omega.
destruct i.
trivial.
eapply IHls.
omega.
Qed.

Lemma removePresent_In :
forall (A : Set)(eqd : eq_dec A)(ls1 ls2 : list A) a,
In a ls2 ->
(~ In a ls1) ->
In a (removePresent eqd ls1 ls2).