intros z1 z2 H H0; case (Z_eq_dec z2 0); intros Z1.
rewrite Z1; red in |- *; case z1; simpl in |- *; auto; intros; red in |- *;
intros; discriminate.
case (ZquotientProp z1 z2); auto; intros r (H1, (H2, H3)).
case (Zle_or_lt 0 (Zquotient z1 z2)); auto; intros Z2.
Contradict H3; apply Zle_not_lt.
replace r with (z1 - Zquotient z1 z2 * z2)%Z;
[ idtac | pattern z1 at 1 in |- *; rewrite H1; ring ].
repeat rewrite Zabs_eq; auto.
pattern z2 at 1 in |- *; replace z2 with (0 + 1 * z2)%Z; [ idtac | ring ].
unfold Zminus in |- *; apply Zle_trans with (z1 + 1 * z2)%Z; auto with zarith.
apply Zplus_le_compat_l.
rewrite Zopp_mult_distr_l.
apply Zle_Zmult_comp_r; auto with zarith.
unfold Zminus in |- *; rewrite Zopp_mult_distr_l; auto with zarith.
Qed.
(* The property of a  number to divide another one (Ndivides n m) shoud be read as m divides n *)
Definition Zdivides (n m : Z) := exists q : Z, n = (m * q)%Z.

Theorem ZdividesZquotient :
forall n m : Z, m <> 0%Z -> Zdivides n m -> n = (Zquotient n m * m)%Z.