+qed.
+
+theorem lt_O_exp: \forall n,m:nat. O < n \to O < n \sup m.
+intros.elim m.simplify.unfold lt.apply le_n.
+simplify.unfold lt.rewrite > times_n_SO.
+apply le_times.assumption.assumption.
+qed.
+
+theorem lt_m_exp_nm: \forall n,m:nat. (S O) < n \to m < n \sup m.
+intros.elim m.simplify.unfold lt.reflexivity.
+simplify.unfold lt.
+apply (trans_le ? ((S(S O))*(S n1))).
+simplify.
+rewrite < plus_n_Sm.apply le_S_S.apply le_S_S.
+rewrite < sym_plus.
+apply le_plus_n.
+apply le_times.assumption.assumption.
+qed.
+
+theorem exp_to_eq_O: \forall n,m:nat. (S O) < n
+\to n \sup m = (S O) \to m = O.
+intros.apply antisym_le.apply le_S_S_to_le.
+rewrite < H1.change with (m < n \sup m).
+apply lt_m_exp_nm.assumption.
+apply le_O_n.
+qed.
+
+theorem injective_exp_r: \forall n:nat. (S O) < n \to
+injective nat nat (\lambda m:nat. n \sup m).
+simplify.intros 4.
+apply (nat_elim2 (\lambda x,y.n \sup x = n \sup y \to x = y)).
+intros.apply sym_eq.apply (exp_to_eq_O n).assumption.
+rewrite < H1.reflexivity.
+intros.apply (exp_to_eq_O n).assumption.assumption.
+intros.apply eq_f.
+apply H1.
+(* esprimere inj_times senza S *)
+cut (\forall a,b:nat.O < n \to n*a=n*b \to a=b).
+apply Hcut.simplify.unfold lt.apply le_S_S_to_le. apply le_S. assumption.
+assumption.
+intros 2.
+apply (nat_case n).
+intros.apply False_ind.apply (not_le_Sn_O O H3).
+intros.
+apply (inj_times_r m1).assumption.
+qed.
+
+variant inj_exp_r: \forall p:nat. (S O) < p \to \forall n,m:nat.
+p \sup n = p \sup m \to n = m \def
+injective_exp_r.