+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: uses: lpxs_pair_refl *)
+lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 →
+ ∀I. ❪G,L1.ⓘ[I]❫ ⊢ ⬈*[h] L2.ⓘ[I].
+/2 width=1 by lex_bind_refl_dx/ qed.
+
+lemma lpxs_pair (h) (G): ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 →
+ ∀V1,V2. ❪G,L1❫ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ❪G,L1.ⓑ[I]V1❫ ⊢ ⬈*[h] L2.ⓑ[I]V2.
+/2 width=1 by lex_pair/ qed.
+
+lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
+/2 width=1 by lex_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: was: lpxs_inv_atom1 *)
+lemma lpxs_inv_atom_sn (h) (G): ∀L2. ❪G,⋆❫ ⊢ ⬈*[h] L2 → L2 = ⋆.
+/2 width=2 by lex_inv_atom_sn/ qed-.
+
+lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ❪G,K1.ⓘ[I1]❫ ⊢ ⬈*[h] L2 →
+ ∃∃I2,K2. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ[I2].
+/2 width=1 by lex_inv_bind_sn/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_pair1 *)
+lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ❪G,K1.ⓑ[I]V1❫ ⊢ ⬈*[h] L2 →
+ ∃∃K2,V2. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ[I]V2.
+/2 width=1 by lex_inv_pair_sn/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_atom2 *)
+lemma lpxs_inv_atom_dx (h) (G): ∀L1. ❪G,L1❫ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
+/2 width=2 by lex_inv_atom_dx/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_pair2 *)
+lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ❪G,L1❫ ⊢ ⬈*[h] K2.ⓑ[I]V2 →
+ ∃∃K1,V1. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ[I]V1.
+/2 width=1 by lex_inv_pair_dx/ qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_2A1: was: lpxs_ind_alt *)
+lemma lpxs_ind (h) (G): ∀Q:relation lenv.
+ Q (⋆) (⋆) → (
+ ∀I,K1,K2.
+ ❪G,K1❫ ⊢ ⬈*[h] K2 →
+ Q K1 K2 → Q (K1.ⓘ[I]) (K2.ⓘ[I])
+ ) → (
+ ∀I,K1,K2,V1,V2.
+ ❪G,K1❫ ⊢ ⬈*[h] K2 → ❪G,K1❫ ⊢ V1 ⬈*[h] V2 →
+ Q K1 K2 → Q (K1.ⓑ[I]V1) (K2.ⓑ[I]V2)
+ ) →
+ ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 → Q L1 L2.
+/3 width=4 by lex_ind/ qed-.