--- /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 "ground/lib/subset_inclusion.ma".
+include "ground/lib/subset_ext.ma".
+
+(* EXTENSIONS FOR SUBSETS ***************************************************)
+
+(* Constructions with subset_inclusion **************************************)
+
+lemma subset_inclusion_ext_f1_bi (A1) (A0) (f) (u1) (v1):
+ u1 ⊆ v1 → subset_ext_f1 A1 A0 f u1 ⊆ subset_ext_f1 A1 A0 f v1.
+#A1 #A0 #f #u1 #v1 #Huv1 #a0 * #a1 #Hau1 #H destruct
+/3 width=3 by subset_in_ext_f1_dx/
+qed.
+
+lemma subset_inclusion_ext_p1_trans (A1) (Q) (u1) (v1):
+ u1 ⊆ v1 → subset_ext_p1 A1 Q v1 → subset_ext_p1 A1 Q u1.
+#A1 #Q #u1 #v1 #Huv1 #Hv1
+/3 width=1 by/
+qed-.