intros l0 r r0 b0 Balanced_l0 height_r.
inversion_clear Balanced_l0.
apply Is_Left_Balanced_Avl_Left_Shift.
rewrite <- height_r; assumption.
apply Is_Fully_Balanced_Avl_Left_Shift.
rewrite <- height_r; assumption.
apply Is_Right_Balanced_Avl_Left_Shift.
rewrite <- H in height_r.
injection height_r; trivial.
Qed.

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