@nlt_i >nplus_succ_dx /2 width=1 by nle_plus_bi_sn/
qed.
-(*** lt_plus_Sn_r *) (**)
-lemma lt_plus_Sn_r: ∀a,x,n. a < a + x + ↑n.
-/2 width=1/ qed-.
+lemma nlt_plus_dx_dx (o) (m) (n): m < n → m < n + o.
+/2 width=1 by nle_plus_dx_dx/ qed.
+
+lemma nlt_plus_dx_sn (o) (m) (n) : m < n → m < o + n.
+/2 width=1 by nle_plus_dx_sn/ qed.
+
+lemma nlt_succ_plus_dx_refl_sn (m) (n): m < ↑(m + n).
+/2 width=1/ qed.
(* Inversions with nplus ****************************************************)