+(* Basic_2A1: was: lpx_sn_inv_pair1 *)
+lemma lex_inv_pair_sn: ∀R,I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R] L2 →
+ ∃∃K2,V2. K1 ⪤[R] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I #L2 #K1 #V1 #H
+elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
+elim (ext2_inv_pair_sn … HZ2) -HZ2 #V2 #HV12 #H destruct
+/2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic_2A1: was: lpx_sn_inv_pair2 *)
+lemma lex_inv_pair_dx: ∀R,I,L1,K2,V2. L1 ⪤[R] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤[R] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I #L1 #K2 #V2 #H
+elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
+elim (ext2_inv_pair_dx … HZ1) -HZ1 #V1 #HV12 #H destruct
+/2 width=5 by ex3_2_intro/
+qed-.
+