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diff --git a/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_fqus.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_2A/multiple/fqus_alt.ma".
+include "basic_2A/multiple/lleq_drop.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on supclosure *************************************************)
+
+lemma lleq_fqu_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐ ⦃G2, K2, U⦄ →
+                      ∀L1. L1 ≡[T, 0] L2 →
+                      ∃∃K1. ⦃G1, L1, T⦄ ⊐ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx … H … I L2 V) -H //
+  #K1 #H1 #H2 lapply (drop_inv_O2 … H1) -H1
+  #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
+| * [ #a ] #I #G #L2 #V #T #L1 #H
+  [ elim (lleq_inv_bind … H)
+  | elim (lleq_inv_flat … H)
+  ] -H
+  /2 width=3 by fqu_pair_sn, ex2_intro/
+| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O … H) -H
+  #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
+| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat … H) -H
+  /2 width=3 by fqu_flat_dx, ex2_intro/
+| #G #L2 #K2 #T #U #m #HLK2 #HTU #L1 #HL12
+  elim (drop_O1_le (Ⓕ) (m+1) L1)
+  [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
+  | lapply (drop_fwd_length_le2 … HLK2) -K2
+    lapply (lleq_fwd_length … HL12) -T -U //
+  ]
+]
+qed-.
+
+lemma lleq_fquq_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮ ⦃G2, K2, U⦄ →
+                       ∀L1. L1 ≡[T, 0] L2 →
+                       ∃∃K1. ⦃G1, L1, T⦄ ⊐⸮ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen … H) -H
+[ #H elim (lleq_fqu_trans … H … HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lleq_fqup_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+ ⦃G2, K2, U⦄ →
+                       ∀L1. L1 ≡[T, 0] L2 →
+                       ∃∃K1. ⦃G1, L1, T⦄ ⊐+ ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans … HTU … HL12) -L2
+  /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU … HL12) -L2
+  #K1 #HTU #HK1 elim (lleq_fqu_trans … HU2 … HK1) -K
+  /3 width=5 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma lleq_fqus_trans: ∀G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐* ⦃G2, K2, U⦄ →
+                       ∀L1. L1 ≡[T, 0] L2 →
+                       ∃∃K1. ⦃G1, L1, T⦄ ⊐* ⦃G2, K1, U⦄ & K1 ≡[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen … H) -H
+[ #H elim (lleq_fqup_trans … H … HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.