--- /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 "static_2/static/lsubr.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties **********************************************************)
+
+theorem lsubr_trans: Transitive … lsubr.
+#L1 #L #H elim H -L1 -L //
+[ #I #L1 #L #_ #IH #X #H elim (lsubr_inv_bind1 … H) -H *
+ [ #L2 #HL2 #H | #L2 #V #W #HL2 #H1 #H2 | #I1 #I2 #L2 #V #Hl2 #H1 #H2 ]
+ destruct /3 width=1 by lsubr_bind, lsubr_beta, lsubr_unit/
+| #L1 #L #V #W #_ #IH #X #H elim (lsubr_inv_abst1 … H) -H *
+ [ #L2 #HL2 #H | #I #L2 #HL2 #H ]
+ destruct /3 width=1 by lsubr_beta, lsubr_unit/
+| #I1 #I2 #L1 #L #V #_ #IH #X #H elim (lsubr_inv_unit1 … H) -H
+ /4 width=1 by lsubr_unit/
+]
+qed-.