cut (up (IZR z) = (z + 1)%Z).
intros Z1; rewrite Z1; rewrite Zplus_comm; apply Zminus_plus;
auto with zarith.
apply sym_equal; apply tech_up; simpl in |- *; auto with real zarith.
replace (IZR z) with (IZR z + IZR 0)%R; try rewrite plus_IZR;
auto with real zarith.
Qed.

Theorem Zlt_Rlt : forall z1 z2 : Z, (IZR z1 < IZR z2)%R -> (z1 < z2)%Z.
intros z1 z2 H; case (Zle_or_lt z2 z1); auto.