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 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/static/lfeq.ma".
+include "basic_2/static/lfeq_fsle.ma".
 include "basic_2/static/lfdeq.ma".
 
 (* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
 
 (* Properties with syntactic equivalence on referred entries ****************)
 
-lemma lfeq_lfdeq: ∀h,o,L1,L2,T. L1 ≐[T] L2 → L1 ≛[h, o, T] L2.
+lemma lfeq_lfdeq: ∀h,o,L1,L2. ∀T:term. L1 ≡[T] L2 → L1 ≛[h, o, T] L2.
 /2 width=3 by lfxs_co/ qed.
+
+lemma lfeq_lfdeq_trans: ∀h,o,L1,L. ∀T:term. L1 ≡[T] L →
+                        ∀L2. L ≛[h, o, T] L2 → L1 ≛[h, o, T] L2.
+/2 width=3 by lfeq_lfxs_trans/ qed-.