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diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_frees.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/multiple/frees.ma".
+include "basic_2/multiple/llpx_sn_alt_rec.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on context-sensitive free variables ***************************)
+
+fact llpx_sn_frees_trans_aux: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
+                              ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                              ∀L1. llpx_sn R l U L1 L2 → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+#R #H1R #H2R #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
+elim (llpx_sn_inv_alt_r … HL12) -HL12 #HL12 #IH
+lapply (drop_fwd_length_lt2 … HLK2) #Hj
+elim (drop_O1_lt (Ⓕ) L1 j) // -Hj -HL12 #I1 #K1 #W1 #HLK1
+elim (IH … HnU HLK1 HLK2) // -IH -HLK2 /5 width=11 by frees_be/
+qed-.
+
+lemma llpx_sn_frees_trans: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
+                           ∀L1,L2,U,l. llpx_sn R l U L1 L2 →
+                           ∀i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
+/2 width=6 by llpx_sn_frees_trans_aux/ qed-.