+(* Properties with generic slicing ******************************************)
+
+(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
+(* Basic_1: includes: csubc_drop_conf_O *)
+(* Basic_2A1: includes: lsubc_drop_O1_trans *)
+lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≡ K2 →
+ ∃∃K1. ⬇*[b, f] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
+#RP #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by lsubc_pair, drops_refl, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+| #L1 #L2 #V #W #A #HV #H1W #H2W #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=8 by drops_refl, lsubc_beta, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.