+lemma lfpr_inv_sort_pair_sn: ∀h,I,G,Y2,L1,V1,s. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡[h, ⋆s] Y2 →
+ ∃∃L2,V2. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_sort_pair_sn/ qed-.
+
+lemma lfpr_inv_sort_pair_dx: ∀h,I,G,Y1,L2,V2,s. ⦃G, Y1⦄ ⊢ ➡[h, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G, L1⦄ ⊢ ➡[h, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_sort_pair_dx/ qed-.
+