intros t; elim t; clear t.
intros.
apply Below_Avl_Nil.
intros k d l ih_l r ih_r b k0 k1 H0 H1.
apply Below_Avl_Node.
apply (less_trans k k0 k1).
apply (inv_below_avl_key k d l r b k0 H0).
assumption.
apply (ih_r k0 k1 (inv_below_avl_right k d l r b k0 H0) H1).
Qed.

Lemma below_below_avl :
forall (t : avl_tree) (key : Int), is_below t key -> is_below_avl t key.