-lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶* U2 →
+lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
+ ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →