-(* Basic_2A1: was just: lpxs_inv_atom2 *)
-lemma lfpxs_inv_atom2: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
-*)
\ No newline at end of file
+(* Basic_2A1: uses: lpxs_inv_atom2 *)
+lemma lfpxs_inv_atom2: ∀h,I,G,L1. ⦃G, L1⦄ ⊢ ⬈*[h, ⓪{I}] ⋆ → L1 = ⋆.
+/2 width=3 by tc_lfxs_inv_atom_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfpxs_fwd_pair_sn: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈*[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, V] L2.
+/2 width=3 by tc_lfxs_fwd_pair_sn/ qed-.
+
+lemma lfpxs_fwd_flat_dx: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈*[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by tc_lfxs_fwd_flat_dx/ qed-.