+
+lemma ldrop_thin: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → L1 [d, e] ≡ L2.
+/2 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma thin_inv_thin1: ∀I,K1,V1,L2,e. K1. ⓑ{I} V1 [0, e] ≡ L2 → 0 < e →
+ K1 [0, e - 1] ≡ L2.
+#I #K1 #V1 #L2 #e * #X #HK1 #HL2 #e
+elim (ltpss_inv_tpss21 … HK1 ?) -HK1 // #K #V #HK1 #_ #H destruct
+lapply (ldrop_inv_ldrop1 … HL2 ?) -HL2 // /2 width=3/
+qed-.