intros t0 t k key below_t0 equiv_del_t0.
inversion_clear equiv_del_t0.
apply lookup_less_below.
intros k0 d0 lookup_k0.
apply lookup_below_less with t0 d0.
elim (equal_dec k0 key).
intros equal_k0.
elimtype False.
apply H with k0 d0; assumption.
intros not_equal_k0.
apply H1; assumption.
assumption.
Qed.

Lemma above_equiv_del_above :
forall (t0 t : avl_tree) (k key : Int),
is_above t0 k -> equiv_del key t0 t -> is_above t k.