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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
index ce56a3154..3392f1e1a 100644
--- 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
@@ -19,37 +19,45 @@ include "basic_2/i_static/tc_lfxs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma tc_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀T. reflexive … (tc_lfxs R T).
+lemma tc_lfxs_refl: ∀R. c_reflexive … R →
+                    ∀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.
+lemma tc_lfxs_pair_refl: ∀R. c_reflexive … R →
+                         ∀L,V1,V2. CTC … 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/
+/3 width=3 by tc_lfxs_step_dx, lfxs_pair_refl, inj/
+qed.
+
+lemma tc_lfxs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (lexs cfull (cext2 R) f) L1 L2 →
+                  L1 ⪤**[R, T] L2.
+#R #L1 #L2 #T #f #Hf #H elim H -L2
+[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
+/5 width=7 by lexs_sdj, tc_lfxs_step_dx, sdj_isid_sn, inj, ex2_intro/
 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
+lemma tc_lfxs_ind_sn: ∀R. c_reflexive … R →
+                      ∀L1,T. ∀Q:predicate …. Q L1 →
+                      (∀L,L2. L1 ⪤**[R, T] L → L ⪤*[R, T] L2 → Q L → Q L2) →
+                      ∀L2. L1 ⪤**[R, T] L2 → Q L2.
+#R #HR #L1 #T #Q #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
+lemma tc_lfxs_ind_dx: ∀R. c_reflexive … R →
+                      ∀L2,T. ∀Q:predicate …. Q L2 →
+                      (∀L1,L. L1 ⪤*[R, T] L → L ⪤**[R, T] L2 → Q L → Q L1) →
+                      ∀L1. L1 ⪤**[R, T] L2 → Q L1.
+#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
 @(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
 qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma tc_lfxs_inv_bind_void: ∀R. (∀L. reflexive … (R L)) →
+lemma tc_lfxs_inv_bind_void: ∀R. c_reflexive … R →
                              ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
                              L1 ⪤**[R, V] L2 ∧ L1.ⓧ ⪤**[R, T] L2.ⓧ.
 #R #HR #p #I #L1 #L2 #V #T #H @(tc_lfxs_ind_sn … HR … H) -L2
@@ -60,7 +68,7 @@ qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma tc_lfxs_fwd_bind_dx_void: ∀R. (∀L. reflexive … (R L)) →
+lemma tc_lfxs_fwd_bind_dx_void: ∀R. c_reflexive … R →
                                 ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
                                 L1.ⓧ ⪤**[R, T] L2.ⓧ.
 #R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind_void … H) -H //