Proof using.
induction l1; intros.
- contradiction.
- simpl in *. intuition.
+ subst. assert (In e2 (l1 ++ l2)). apply in_or_app. intuition.
find_apply_hyp_hyp. intuition.
+ eauto.
Qed.

(* Specifications: *)
Lemma sorted_app_In_reduce:
forall l1 l2 e1 e2,
sorted (l1 ++ [e1]) ->
sorted (l2 ++ [e2]) ->
eIndex e1 = eIndex e2 ->
(forall e, In e (l1 ++ [e1]) -> In e (l2 ++ [e2])) ->
(forall e, In e l1 -> In e l2).