fact lreq_lpxs_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
                               ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
                               ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
                                    (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
#h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
[ #l #k #_ #L2 #H >(lpxs_inv_atom1 … H) -H
  /3 width=5 by ex3_intro, conj/
| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
| #I #L1 #L0 #V1 #k #HL10 #IHL10 #Hm #Y #H
  elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
  lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
  elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
  @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_pair/
  #T elim (IH T) #HL0dx #HL0sn
  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
| #I1 #I0 #L1 #L0 #V1 #V0 #l #k #HL10 #IHL10 #Hm #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, lreq_succ/
  #T elim (IH T) #HL0dx #HL0sn
  @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
]
qed-.

lemma lreq_lpxs_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
                            ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
                            ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
                                 (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
/2 width=1 by lreq_lpxs_trans_lleq_aux/ qed-.