intros z n H' H'0; case (le_or_lt (Zabs_nat z) n); auto; intros lt.
case (Zle_or_lt 0 z); intros Zle0.
Contradict H'0.
apply Zlt_not_le; auto.
rewrite <- (inj_abs z); auto with zarith.
Contradict H'.
apply Zlt_not_le; auto.
replace z with (- Z_of_nat (Zabs_nat z))%Z.
apply Zlt_Zopp; auto with zarith.
rewrite <- absolu_Zopp.
rewrite inj_abs; auto with zarith.
Qed.

Theorem lt_Zlt_inv : forall n m : nat, (Z_of_nat n < Z_of_nat m)%Z -> n < m.
intros n m H'; case (le_or_lt n m); auto.