diff --git a/src/subsequences.v b/src/subsequences.v
index ae97f2c..b1a28a1 100644
--- a/src/subsequences.v
+++ b/src/subsequences.v
@@ -742,8 +742,7 @@ Proof.
   assert (subsequence ((filter f x) ++ (filter f x0))
                       ((filter f [a]) ++ (filter f v))).
   apply subsequence_app. simpl. rewrite H0.
-  apply subsequence_incl in H1. apply incl_cons_inv in H1.
-  destruct H1.
+  apply subsequence_incl in H1. apply incl_cons_inv in H1. destruct H1.
   assert (In a (filter f x)). apply filter_In. split; assumption.
   apply subsequence_in. assumption. apply IHv. assumption.
   rewrite <- filter_app in H3. rewrite <- H in H3. simpl in H3.
@@ -752,3 +751,25 @@ Proof.
   assumption.
 Qed.
 
+
+Theorem subsequence_add {X: Type} :
+  forall (u v w: list X) x, subsequence u v -> Add x u w -> subsequence w v.
+Proof.
+  intros u v w. generalize u v. induction w; intros u0 v0 x; intro H.
+  intro I. inversion I. intro I.  inversion I.
+  rewrite <- H2. apply subsequence_cons_l. rewrite <- H3. assumption.
+
+  assert (J := H). apply subsequence_eq_def_1 in H. rewrite <- H1 in H.
+  rewrite H0 in H. destruct H. destruct H.
+  destruct x1. apply O_S in H. contradiction. destruct b.
+  rewrite H4. simpl. apply subsequence_double_cons.
+  apply IHw with (u := l) (x := x).
+  apply subsequence_eq_def_3. apply subsequence_eq_def_2. exists x1.
+  inversion H. split; reflexivity. assumption.
+  apply subsequence_cons_r with (a:=a).
+  rewrite H4. simpl. apply subsequence_double_cons.
+  apply IHw with (u := l) (x := x).
+  apply subsequence_eq_def_3. apply subsequence_eq_def_2. exists x1.
+  inversion H. split; reflexivity. assumption.
+Qed.
+