+lemma lexs_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+ ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ∥ g →
+ ∃∃L. L1 ⪤*[RP, R1, g] L & L ⪤*[R2, cfull, f] L2.
+#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=3 by lexs_atom, ex2_intro/ ]
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
+[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
+]
+qed-.
+