intros; elim q.
(* O *)
unfold posOZ in |- *; rewrite (mult_OZ (pos n)).
cut (IZ = pos 0); intros. elim H. rewrite (mult_IZ (pos n)).
elim (addZ_opposite (pos n) I); intros; elim H1; intros; elim H3; intros.
rewrite H5; reflexivity. reflexivity.
(* S y *)
intros y H; unfold posOZ in |- *;
elim (multZ_commutativity (pos (S y)) (pos n)).
rewrite (tech_mult_posZ y (pos n)).
rewrite
(opp_add Z IdZ addZ OZ oppZ Z_group addZ_commutativity
(multZ (pos y) (pos n)) (pos n) I I).
elim
(addZ_associativity (oppZ (multZ (pos y) (pos n))) (oppZ (pos n)) (pos n)).
elim (addZ_opposite (pos n) I); intros; elim H1; intros; elim H3; intros.
rewrite H5; rewrite (add_OZ (oppZ (multZ (pos y) (pos n)))).
elim (multZ_commutativity (pos y) (pos n)); reflexivity.
Qed.

(***************)
Lemma tech_div32 :
forall n q r : nat, S n > r -> pos (n - r) = addZ (pos n) (oppZ (posOZ r)).