+
+fact leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ⩬[d, e] L0 → e = ∞ →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃L. L ⩬[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+ (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L).
+#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
+[ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
+ /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
+| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ lapply (ysucc_inv_Y_dx … He) -He #He
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
+]
+qed-.
+
+lemma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ⩬[d, ∞] L0 →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃L. L ⩬[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+ (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L).
+/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.