induction ls1; intuition; simpl in *.
intuition; subst.
eapply (IHls1 (removeFirst eqd ls2 a)).
eapply H3.
intuition.
eapply removeFirst_not_in.
intuition.
eapply H0.
eauto.
eapply H.
eauto.
Qed.

(* Specifications: *)
Lemma matchOrder_not_in : forall (A : Set)(eqd : eq_dec A)(ls1 ls2 : list A) a,
(forall a, In a ls1 -> In a ls2) ->
~In a ls2 ->
~In a (matchOrder eqd ls1 ls2).