--- /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_2/relocation/rtmap_id.ma".
+include "static_2/notation/relations/subseteq_4.ma".
+include "static_2/syntax/lveq.ma".
+include "static_2/static/frees.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+definition fsle: bi_relation lenv term ≝ λL1,T1,L2,T2.
+ ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 &
+ L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+
+interpretation "free variables inclusion (restricted closure)"
+ 'SubSetEq L1 T1 L2 T2 = (fsle L1 T1 L2 T2).
+
+interpretation "free variables inclusion (term)"
+ 'subseteq T1 T2 = (fsle LAtom T1 LAtom T2).
+
+(* Basic properties *********************************************************)
+
+lemma fsle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄.
+/3 width=8 by frees_sort, sle_refl, ex4_4_intro/ qed.
+
+lemma fsle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄.
+/3 width=8 by frees_gref, sle_refl, ex4_4_intro/ qed.