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diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lcoeq/lcoeq_lcoeq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lcoeq/lcoeq_lcoeq.etc
new file mode 100644
index 000000000..29777537e
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+++ b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lcoeq/lcoeq_lcoeq.etc
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/grammar/lcoeq.ma".
+
+(* COEQUIVALENCE FOR LOCAL ENVIRONMENTS *************************************)
+
+(* Main properties **********************************************************)
+
+theorem lcoeq_trans: ∀d,e. Transitive … (lcoeq d e).
+#d #e #L1 #L2 #H elim H -L1 -L2 -d -e //
+[ #I #L1 #L #V #HL1 #_ #X #H >(lcoeq_inv_O2 … HL1) -HL1 //
+| #I1 #I #L1 #L #V1 #V #e #_ #IHL1 #X #H elim (lcoeq_inv_pair1 … H) -H //
+  #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lcoeq_pair/
+| #I #L1 #L #V #d #e #_ #IHL1 #X #H elim (lcoeq_inv_succ1 … H) -H //
+  #L2 #HL2 #H destruct /3 width=1 by lcoeq_succ/
+]
+qed-.