--- /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/syntax/acle.ma".
+
+(* APPLICABILITY CONDITION PREORDER *****************************************)
+
+(* Main properties **********************************************************)
+
+theorem acle_trans: Transitive … acle.
+#a1 #a #Ha1 #a2 #Ha2 #m1 #Hm1
+elim (Ha1 … Hm1) -Ha1 -Hm1 #m #Ha #Hm1
+elim (Ha2 … Ha) -Ha2 -Ha #m2 #Ha2 #Hm2
+/3 width=5 by transitive_le, ex2_intro/
+qed-.