--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: Transitive … lsuby.
+#L1 #L #H elim H -L1 -L
+[ #L1 #X #H lapply (lsuby_inv_atom1 … H) -H //
+| #I1 #I #L1 #L #V #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H // *
+ #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+]
+qed-.