-lemma csx_fquq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬈*[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
-[ /2 width=5 by csx_fqu_conf/
-| * #HG #HL #HT destruct //
+(* Properties with extended supclosure **************************************)
+
+lemma csx_fqu_conf: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by csx_inv_lref_pair, drops_refl/
+| /2 width=3 by csx_fwd_pair_sn/
+| /2 width=2 by csx_fwd_bind_dx/
+| /2 width=4 by csx_fwd_bind_dx_unit/
+| /2 width=3 by csx_fwd_flat_dx/
+| /4 width=7 by csx_inv_lifts, drops_refl, drops_drop/