+lemma frees_lexs_conf: ∀R. lfxs_fle_compatible R →
+ ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
+ ∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 →
+ ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
+#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
+lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
+@(fle_frees_trans_eq … H2L … Hf1) /3 width=4 by lexs_fwd_length, sym_eq/
+qed-.
+
+(* Properties with free variables inclusion for restricted closures *********)
+
+(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
+lemma fle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
+ ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
+#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
+elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
+/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lfxs_sym: ∀R. lfxs_fle_compatible R →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (lfxs R T).
+#R #H1R #H2R #T #L1 #L2
+* #f1 #Hf1 #HL12
+elim (frees_lexs_conf … Hf1 … HL12) -Hf1 //
+/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
+qed-.
+