--- /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/syntax/lveq_length.ma".
+include "basic_2/static/fsle.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Properties with length for local environments ****************************)
+
+lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
+/3 width=8 by lveq_length_eq, frees_sort, sle_refl, ex4_4_intro/ qed.
+
+lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
+/3 width=8 by lveq_length_eq, frees_gref, sle_refl, ex4_4_intro/ qed.
+
+lemma fsle_zero_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
+ ∀I1,I2. ⦃K1.ⓑ{I1}V1, #O⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+#K1 #K2 #HK #V1 #V2
+* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
+#I1 #I2
+elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
+/3 width=12 by frees_pair, lveq_bind, sle_next, ex4_4_intro/
+qed.