intros m n q r H' H'0 H'1 H'2.
case (ZquotientProp m n); auto; intros z (H0, (H1, H2)).
case (Zle_or_lt (Zabs q) (Zabs (Zquotient m n))); intros Zl1; auto with arith.
case (Zle_lt_or_eq _ _ Zl1); clear Zl1; intros Zl1; auto with arith.
Contradict H1; apply Zlt_not_le.
pattern m at 1 in |- *; rewrite H'0.
apply Zle_lt_trans with (Zabs (q * n) + Zabs r)%Z; auto with zarith.
apply Zlt_le_trans with (Zabs (q * n) + Zabs n)%Z; auto with zarith.
repeat rewrite Zabs_Zmult.
replace (Zabs q * Zabs n + Zabs n)%Z with (Zsucc (Zabs q) * Zabs n)%Z;
[ auto with zarith | unfold Zsucc in |- *; ring ].
case (Zabs_eq_case _ _ Zl1); auto.
intros H;
(cut (Zquotient m n = 0%Z);
[ intros H3; rewrite H; repeat rewrite H3; simpl in |- *; auto | idtac ]).
cut (Zabs (Zquotient m n) < 1)%Z.
case (Zquotient m n); simpl in |- *; auto; intros p; case p;
unfold Zlt in |- *; simpl in |- *; intros; discriminate.
apply Zlt_mult_simpl_l with (c := Zabs n); auto with zarith.
case (Zle_lt_or_eq 0 (Zabs n)); auto with zarith.
intros H3; case H'; auto.
generalize H3; case n; simpl in |- *; auto; intros; discriminate.
rewrite <- Zabs_Zmult; rewrite (Zmult_comm n).
replace (Zabs n * 1)%Z with (Zabs n); [ idtac | ring ].
apply Zle_lt_trans with (1 := H1).
apply Zlt_mult_simpl_l with (c := (1 + 1)%Z); auto with zarith.
replace ((1 + 1) * Zabs m)%Z with (Zabs (m + m)).
replace ((1 + 1) * Zabs n)%Z with (Zabs n + Zabs n)%Z; [ idtac | ring ].
pattern m at 1 in |- *; rewrite H'0; rewrite H0; rewrite H.
replace (- Zquotient m n * n + r + (Zquotient m n * n + z))%Z with (r + z)%Z;
[ idtac | ring ].
apply Zle_lt_trans with (Zabs r + Zabs z)%Z; auto with zarith.
rewrite <- (Zabs_eq (1 + 1)); auto with zarith.
rewrite <- Zabs_Zmult; apply f_equal with (f := Zabs); auto with zarith.
Contradict H'1; apply Zlt_not_le.
pattern m at 1 in |- *; rewrite H0.
apply Zle_lt_trans with (Zabs (Zquotient m n * n) + Zabs z)%Z;
auto with zarith.
apply Zlt_le_trans with (Zabs (Zquotient m n * n) + Zabs n)%Z;
auto with zarith.
repeat rewrite Zabs_Zmult.
replace (Zabs (Zquotient m n) * Zabs n + Zabs n)%Z with
(Zsucc (Zabs (Zquotient m n)) * Zabs n)%Z;
[ auto with zarith | unfold Zsucc in |- *; ring ].
Qed.

Theorem ZquotientZopp :
forall m n : Z, Zquotient (- m) n = (- Zquotient m n)%Z.