intros t avl_t.
elim avl_t; clear avl_t t.
intros; apply Below_Nil.
intros k d l r b avl_l ih_l avl_r ih_r Balanced_t below_l above_r k0
below_avl.
inversion_clear below_avl.
apply Below_Node.
assumption.
apply ih_l.
apply below_avl_trans with k.
assumption.
assumption.
apply ih_r; assumption.
Qed.

Lemma is_above_avl_is_above :
forall t : avl_tree,
is_avl t -> forall k0 : Int, is_above_avl t k0 -> is_above t k0.