(* O *)
reflexivity.
(* S y *)
intros; rewrite (tech_add_pos_succZ n0).
elim succ_addZ_r; exact H.
Qed.

(*********************)
Lemma tech_add_neg_posZ : forall n : nat, addZ (neg n) (pos n) = OZ.
intros; elim (addZ_commutativity (pos n) (neg n));
exact (tech_add_pos_negZ n).