-lemma fle_inv_voids_sn_frees_dx: ∀L1,L2,T1,T2,n. ⦃ⓧ*[n]L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- |L1| = |L2| → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
- ∃∃f1. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & ⫱*[n]f1 ⊆ f2.
-#L1 #L2 #T1 #T2 #n #H #HL12 #f2 #Hf2
-elim (fle_inv_voids_sn … H HL12) -H -HL12 #f1 #g2 #Hf1 #Hg2 #Hfg
-lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hfg2
-lapply (sle_eq_repl_back2 … Hfg … Hfg2) -g2
+lemma fle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
+ ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 &
+ L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
+lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
+lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
+lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
+/2 width=6 by ex3_3_intro/
+qed-.
+
+lemma fle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
+ ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
+ ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & f1 ⊆ f2.
+#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
+elim (fle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
+elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct