+
+(* Properties on supclosure *************************************************)
+
+lemma fsup_tpss_trans_full: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶*[0,|L2|] U2 →
+ ∃∃L,U1. L1 ⊢ ▶*[0,|L1|] L & L ⊢ T1 ▶*[0,|L|] U1 & ⦃L, U1⦄ ⊃ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
+#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
+elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (ltpss_sn_fwd_length … HK1) #H >H in HK1; -H #HK1
+elim (le_or_ge d (|K|)) #Hd
+[ elim (ldrop_ltpss_sn_trans_be … HLK1 … HK1 … Hd) // -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tpss_lift_be … HT1 … Hd HL2K HTU1 … HTU) // -HT1 -HTU1 #HU1
+| elim (ldrop_ltpss_sn_trans_ge … HLK1 … HK1 Hd) -HLK1 -HK1 #L2 #HL12 #HL2K
+ lapply (tpss_lift_le … HT1 … Hd HL2K HTU1 … HTU) -HT1 -HTU1 #HU1
+]
+lapply (ltpss_sn_weak_full … HL12) -HL12 #HL12
+lapply (tpss_weak_full … HU1) -HU1 #HU1
+@(ex3_2_intro … L2 U) // /2 width=7/ (**) (* explicit constructor: auto /3 width=14/ too slow *)
+qed-.