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diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
index 0202dc926..89ef40e91 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/substitution/lift_constructors.ma
@@ -12,26 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "delayed_updating/substitution/lift_eq.ma".
-
-axiom lift_path_after (p) (f1) (f2):
-      ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-(*
-#p @(path_ind_lift … p) -p // [ #n | #n #l #p | #p ] #IH #f1 #f2
-[ <lift_path_d_empty_sn <lift_path_d_empty_sn
-| <lift_path_d_lcons_sn <lift_path_d_lcons_sn
-| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
-*)
-include "delayed_updating/substitution/lift_prototerm.ma".
-
-axiom lift_term_after (t) (f1) (f2):
-      ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
-
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
 include "delayed_updating/syntax/prototerm_constructors.ma".
 
 (* LIFT FOR PROTOTERM *******************************************************)
 
-lemma lift_iref_after_sn (f) (t) (n:pnat):
+lemma lift_iref_after_sn (f) (t:prototerm) (n:pnat):
       ↑[f∘𝐮❨n❩]t ⊆ ↑[f](𝛗n.t).
 #f #t #n #p * #q #Hq #H0 destruct
 @(ex2_intro … (𝗱n◗𝗺◗q))