Proof with auto.
intros z X u e H.
rewrite subst_ee_open_te...
Qed.

(* Specifications: *)
Lemma subst_ee_intro_rec : forall x e u k,
x `notin` fv_ee e ->
open_ee_rec k u e = subst_ee x u (open_ee_rec k (exp_fvar x) e).