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diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_after_ist_isi.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_after_ist_isi.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground/relocation/gr_ist_isi.ma".
+include "ground/relocation/gr_after_ist.ma".
+
+(* RELATIONAL COMPOSITION FOR GENERIC RELOCATION MAPS ***********************************************************)
+
+(* Forward lemmas on istot and isid **************************************************)
+
+(*** after_fwd_isid_sn *)
+lemma gr_after_des_isi_sn:
+      ∀f2,f1,f. 𝐓❪f❫ → f2 ⊚ f1 ≘ f → f1 ≡ f → 𝐈❪f2❫.
+#f2 #f1 #f #H #Hf elim (gr_after_inv_ist … Hf H) -H
+#Hf2 #Hf1 #H @gr_isi_pat_total // -Hf2
+#i2 #i #Hf2 elim (Hf1 i2) -Hf1
+#i0 #Hf1 lapply (gr_pat_increasing … Hf1)
+#Hi20 lapply (gr_after_des_pat_sn … i0 … Hf1 … Hf) -Hf
+/3 width=7 by gr_pat_eq_repl_back, gr_pat_mono, gr_pat_id_le/
+qed-.
+
+(*** after_fwd_isid_dx *)
+lemma gr_after_des_isi_dx:
+      ∀f2,f1,f.  𝐓❪f❫ → f2 ⊚ f1 ≘ f → f2 ≡ f → 𝐈❪f1❫.
+#f2 #f1 #f #H #Hf elim (gr_after_inv_ist … Hf H) -H
+#Hf2 #Hf1 #H2 @gr_isi_pat_total // -Hf1
+#i1 #i2 #Hi12 elim (gr_after_pat_sn_des … Hi12 … Hf) -f1
+/3 width=8 by gr_pat_inj, gr_pat_eq_repl_back/
+qed-.