intros z H'; apply inject_nat_eq.
rewrite inj_S.
repeat rewrite <- (absolu_Zopp (Zsucc z)).
repeat rewrite <- (absolu_Zopp z).
repeat rewrite inj_abs; replace 0%Z with (- (0))%Z; auto with zarith.
Qed.

Theorem Zlt_absolu :
forall (x : Z) (n : nat), Zabs_nat x < n -> (x < Z_of_nat n)%Z.