+elim (rpx_fwd_lpx_req … H) -H #K1 #HLK1 #HK12
+/3 width=3 by req_reqx, ex2_intro/
+qed-.
+
+(* Inversion lemmas with sort-irrelevant equivalence for local environments *)
+
+lemma rpx_inv_reqx_lpx (G) (T):
+ ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T] L2 →
+ ∃∃L. L1 ≛[T] L & ❪G,L❫ ⊢ ⬈ L2.
+#G #T #L1 #L2 #H
+elim (rpx_inv_req_lpx … H) -H #L #HL1 #HL2
+/3 width=3 by req_reqx, ex2_intro/
+qed-.
+
+(* Forward lemmas with sort-irrelevant equivalence for local environments ***)
+
+lemma rpx_fwd_lpx_reqx (G) (T):
+ ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T] L2 →
+ ∃∃L. ❪G,L1❫ ⊢ ⬈ L & L ≛[T] L2.
+#G #T #L1 #L2 #H
+elim (rpx_fwd_lpx_req … H) -H #L #HL1 #HL2