-lemma fle_bind_dx: ∀T,U. ⬆*[1] T ≡ U →
- ∀p,I,L,V. ⦃L, T⦄ ⊆ ⦃L, ⓑ{p,I}V.U⦄.
-#T #U #HTU #p #I #L #V
-elim (frees_total L V) #f1 #Hf1
-elim (frees_total L T) #f2 #Hf2
-elim (sor_isfin_ex f1 f2) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #H #_
-lapply (sor_inv_sle_dx … H) #Hf0
->(tl_push_rew f) in H; #Hf
-/6 width=6 by frees_lifts_SO, frees_bind, drops_refl, drops_drop, ex3_2_intro/
+lemma fle_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≡ U1 → ∀L1,L2. |L2| ≤ |L1| →
+ ∀T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ⦃L1.ⓧ, U1⦄ ⊆ ⦃L2, T2⦄.
+#T1 #U1 #HTU1 #L1 #L2 #H1L #T2
+* #n #m #f #g #Hf #Hg #H2L #Hfg
+lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct
+lapply (frees_lifts_SO (Ⓣ) (L1.ⓧ) … HTU1 … Hf)
+[ /3 width=4 by drops_refl, drops_drop/ ] -T1 #Hf
+@(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_sn/ (**) (* explict constructor *)
+qed-.
+
+lemma fle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ →
+ ∀U1,U2. ⬆*[1] T1 ≡ U1 → ⬆*[1] T2 ≡ U2 →
+ ∀I1,I2. ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄.
+#K1 #K2 #HK #T1 #T2
+* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
+#U1 #U2 #HTU1 #HTU2 #I1 #I2
+elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
+/5 width=12 by frees_lifts_SO, drops_refl, drops_drop, lveq_bind, sle_push, ex4_4_intro/