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diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_coafter_nat_tls_pushs.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_coafter_nat_tls_pushs.ma
new file mode 100644
index 000000000..9445dccd0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/pr_coafter_nat_tls_pushs.ma
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/pr_pushs.ma".
+include "ground/relocation/pr_tls.ma".
+include "ground/relocation/pr_nat.ma".
+include "ground/relocation/pr_coafter.ma".
+
+(* RELATIONAL CO-COMPOSITION FOR PARTIAL RELOCATION MAPS ********************)
+
+(* Destructions with pr_nat and pr_tls and pr_pushs *************************)
+
+(*** coafter_fwd_pushs *)
+lemma pr_coafter_des_pushs_dx (n) (m):
+      ∀g2,f1,g. g2 ~⊚ ⫯*[m]f1 ≘ g → @↑❪m, g2❫ ≘ n →
+      ∃∃f. ⫰*[n]g2 ~⊚ f1 ≘ f & ⫯*[n] f = g.
+#n @(nat_ind_succ … n) -n
+[ #m #g2 #f1 #g #Hg #H
+  elim (pr_nat_inv_zero_dx … H) -H [|*: // ] #f2 #H1 #H2 destruct
+  /2 width=3 by ex2_intro/
+| #n #IH * [| #m ] #g2 #f1 #g #Hg #H
+  [ elim (pr_nat_inv_zero_succ … H) -H [|*: // ] #f2 #Hmn #H destruct
+    elim (pr_coafter_inv_next_sn … Hg) -Hg [|*: // ] #f #Hf #H destruct
+    elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
+  | elim (pr_nat_inv_succ_bi … H) -H [1,4: * |*: // ] #f2 #Hmn #H destruct
+    [ elim (pr_coafter_inv_push_bi … Hg) -Hg [|*: // ] #f #Hf #H destruct
+      elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
+    | elim (pr_coafter_inv_next_sn … Hg) -Hg [|*: // ] #f #Hf #H destruct
+      elim (IH … Hf Hmn) -IH -Hf -Hmn /2 width=3 by ex2_intro/
+    ]
+  ]
+]
+qed-.