Proof with auto.
induction e1; intros e2 Z P k; simpl; f_equal...
Case "exp_bvar".
destruct (k === n)...
Qed.

(* Specifications: *)
Lemma subst_te_open_ee : forall e1 e2 Z P,
subst_te Z P (open_ee e1 e2) = open_ee (subst_te Z P e1) (subst_te Z P e2).