intros key update init t0 t k0 equiv_t0 less below_t0.
apply lookup_less_below.
intros k d lookup_t.
elim (equal_dec k key).
intro equal.
apply (equal_less_less k key k0); assumption.
intro notequal.
generalize
(inv_equiv_ins_notequal1 key update init t0 t k d equiv_t0 notequal lookup_t).
intro lookup_t0.
apply (lookup_below_less k t0 d); assumption.
Qed.

Lemma equiv_ins_above :
forall (key : Int) (update : B -> B) (init : B) (t0 t : avl_tree) (k0 : Int),
equiv_ins key update init t0 t ->
Less k0 key -> is_above t0 k0 -> is_above t k0.