cut (up (INR n) = (Z_of_nat n + Z_of_nat 1)%Z).
intros H'; rewrite H'; simpl in |- *; ring.
apply sym_equal; apply tech_up; auto.
replace (Z_of_nat n + Z_of_nat 1)%Z with (Z_of_nat (S n)).
repeat rewrite <- INR_IZR_INZ.
apply INR_lt_nm; auto.
rewrite Zplus_comm; rewrite <- inj_plus; simpl in |- *; auto.
rewrite plus_IZR; simpl in |- *; auto with real.
repeat rewrite <- INR_IZR_INZ; auto with real.
Qed.

Theorem Int_part_IZR : forall z : Z, Int_part (IZR z) = z.
intros z; unfold Int_part in |- *.