induction ls1; intuition; simpl in *.
destruct (eqd a a0); subst;
intuition.
eapply IHls1.
intuition.
eapply H0.
assert (~ In a0 ls1).
intuition.
specialize (@matchOrder_not_in_h _ eqd ls1 (removeFirst eqd ls2 a) a0); intuition.
exfalso.
eapply H4; eauto.
intuition.
eapply removeFirst_not_in.
intuition.
eapply H0.
eapply H5.
eapply H3.
Qed.

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