intros n m q H; case (Zle_lt_or_eq _ _ H); intros Z0.
case (Z_eq_dec q 0); intros Z1.
rewrite Z1; unfold Zquotient in |- *; case n; case m; simpl in |- *;
auto with zarith.
case (Zle_or_lt (Zabs (Zquotient n q)) (Zabs (Zquotient m q))); auto;
intros H'1.
case (ZquotientProp m q); auto; intros r1 (H'2, (H'3, H'4)); auto with zarith.
case (ZquotientProp n q); auto; intros r2 (H'5, (H'6, H'7)); auto with zarith.
Contradict H'6.
apply Zlt_not_le.
apply Zlt_le_trans with (1 := Z0).
rewrite H'2.
apply Zle_trans with (Zabs (Zquotient m q * q) + Zabs r1)%Z; auto with zarith.
apply Zle_trans with (Zabs (Zquotient m q * q) + Zabs q)%Z; auto with zarith.
repeat rewrite Zabs_Zmult.
replace (Zabs (Zquotient m q) * Zabs q + Zabs q)%Z with
(Zsucc (Zabs (Zquotient m q)) * Zabs q)%Z;
[ idtac | unfold Zsucc in |- *; ring ].
cut (0 < Zabs q)%Z; auto with zarith.
case (Zle_lt_or_eq 0 (Zabs q)); auto with zarith.
intros H'6; case Z1; auto.
generalize H'6; case q; simpl in |- *; auto; intros; discriminate.
case (Zabs_eq_case _ _ Z0); intros Z1; rewrite Z1; auto with zarith.
rewrite ZquotientZopp; rewrite Zabs_Zopp; auto with zarith.
Qed.

Theorem NotDividesDigit :
forall r v : Z,
(1 < r)%Z -> v <> 0%Z -> ~ Zdivides v (Zpower_nat r (digit r v)).