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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_rdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_rdeq.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/rdeq_fqus.ma".
+include "basic_2/rt_transition/cpx_rdeq.ma".
+include "basic_2/rt_transition/lpx_rdeq.ma".
+include "basic_2/rt_transition/fpb.ma".
+
+(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+lemma tdeq_fpb_trans: ∀h,o,U2,U1. U2 ≛[h, o] U1 →
+                      ∀G1,G2,L1,L2,T1. ⦃G1, L1, U1⦄ ≻[h, o] ⦃G2, L2, T1⦄ → 
+                      ∃∃L,T2. ⦃G1, L1, U2⦄ ≻[h, o] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1
+[ #G2 #L2 #T1 #H
+  elim (tdeq_fqu_trans … H … HU21) -H
+  /3 width=5 by fpb_fqu, ex3_2_intro/
+| #T1 #HUT1 #HnUT1
+  elim (tdeq_cpx_trans … HU21 … HUT1) -HUT1
+  /6 width=5 by fpb_cpx, tdeq_canc_sn, tdeq_trans, ex3_2_intro/
+| /6 width=5 by fpb_lpx, rpx_tdeq_div, tdeq_rdeq_conf, ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was just: lleq_fpb_trans *)
+lemma rdeq_fpb_trans: ∀h,o,F,K1,K2,T. K1 ≛[h, o, T] K2 →
+                      ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, o] ⦃G, L2, U⦄ →
+                      ∃∃L1,U0. ⦃F, K1, T⦄ ≻[h, o] ⦃G, L1, U0⦄ & U0 ≛[h, o] U & L1 ≛[h, o, U] L2.
+#h #o #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
+[ #G #L2 #U #H2 elim (rdeq_fqu_trans … H2 … HT) -K2
+  /3 width=5 by fpb_fqu, ex3_2_intro/
+| #U #HTU #HnTU elim (rdeq_cpx_trans … HT … HTU) -HTU
+  /5 width=10 by fpb_cpx, cpx_rdeq_conf_sn, tdeq_trans, tdeq_rdeq_conf, ex3_2_intro/
+| #L2 #HKL2 #HnKL2 elim (rdeq_lpx_trans … HKL2 … HT) -HKL2
+  /6 width=5 by fpb_lpx, (* 2x *) rdeq_canc_sn, ex3_2_intro/
+]
+qed-.