+lemma fpbs_cpxs_tdeq_fqup_lfpx_trans: ∀h,o,G1,G3,L1,L3,T1,T3. ⦃G1, L1, T1⦄ ≥ [h, o] ⦃G3, L3, T3⦄ →
+ ∀T4. ⦃G3, L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛[h, o] T5 →
+ ∀G2,L4,T2. ⦃G3, L3, T5⦄ ⊐+ ⦃G2, L4, T2⦄ →
+ ∀L2. ⦃G2, L4⦄ ⊢ ⬈[h, T2] L2 → ⦃G1, L1, T1⦄ ≥ [h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
+@(fpbs_lfpx_trans … HL42) -L2 (**) (* full auto too slow *)
+@(fpbs_fqup_trans … H34) -G2 -L4 -T2
+/3 width=3 by fpbs_cpxs_trans, fpbs_tdeq_trans/
+qed-.
+