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diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_fqup.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/static/lfxs_fqup.ma".
+include "basic_2/i_static/tc_lfxs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Advanced properties ******************************************************)
+
+lemma tc_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀T. reflexive … (tc_lfxs R T).
+/3 width=1 by lfxs_refl, inj/ qed.
+
+(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
+lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
+                    ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⦻**[R, T] L.ⓑ{I}V2.
+#R #HR #L #V1 #V2 #H elim H -V2
+/3 width=3 by tc_lfxs_step_dx, lfxs_pair, inj/
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma tc_lfxs_ind_sn: ∀R. (∀L. reflexive … (R L)) →
+                      ∀L1,T. ∀R0:predicate …. R0 L1 →
+                      (∀L,L2. L1 ⦻**[R, T] L → L ⦻*[R, T] L2 → R0 L → R0 L2) →
+                      ∀L2. L1 ⦻**[R, T] L2 → R0 L2.
+#R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
+qed-.
+
+lemma tc_lfxs_ind_dx: ∀R. (∀L. reflexive … (R L)) →
+                      ∀L2,T. ∀R0:predicate …. R0 L2 →
+                      (∀L1,L. L1 ⦻*[R, T] L → L ⦻**[R, T] L2 → R0 L → R0 L1) →
+                      ∀L1. L1 ⦻**[R, T] L2 → R0 L1.
+#R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
+qed-.