induction ls; intuition; simpl in *.
inversion H; clear H; subst.
destruct (eqd a0 a); subst.
trivial.
econstructor.
eapply removeFirst_not_in.
intuition.
eapply IHls; eauto.
Qed.

(* Specifications: *)
Lemma removeFirst_in : forall (A : Set)(eqd : eq_dec A)(ls : list A)(a1 a2 : A),
In a1 ls ->
a1 <> a2 ->
In a1 (removeFirst eqd ls a2).