lemma fqu_lpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
                     ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
                     ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/
[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H
  #K2 #W2 #HLK2 #HVW2 #H destruct
  /3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #K2 #HK12
  elim (drop_lpx_trans … HLK1 … HK12) -HK12
  /3 width=7 by fqu_drop, ex3_2_intro/
]
qed-.

lemma fquq_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
                      ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h, o] K2 →
                      ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.