diff --git a/Map.agda b/Map.agda
index 77c9b07..1bdce79 100644
--- a/Map.agda
+++ b/Map.agda
@@ -329,6 +329,22 @@ private module ImplInsert (f : B → B → B) where
                                  Unique (keys l₂) → Unique (keys (intersect l₁ l₂))
     intersect-preserves-Unique {l₁} u = restrict-preserves-Unique (updates-preserve-Unique {l₁} u)
 
+    restrict-needs-both : ∀ {k : A} {v : B} {l₁ l₂ : List (A × B)} →
+                          (k , v) ∈ restrict l₁ l₂ → (k ∈k l₁ × (k , v) ∈ l₂)
+    restrict-needs-both {k} {v} {l₁} {[]} ()
+    restrict-needs-both {k} {v} {l₁} {(k' , v') ∷ xs} k,v∈l₁l₂
+        with ∈k-dec k' l₁ | k,v∈l₁l₂
+    ...   | yes k'∈kl₁    | here k,v≡k',v'
+            rewrite cong proj₁ k,v≡k',v' rewrite cong proj₂ k,v≡k',v' =
+            (k'∈kl₁ , here refl)
+    ...   | yes _         | there k,v∈l₁xs =
+            let (k∈kl₁ , k,v∈xs) = restrict-needs-both k,v∈l₁xs
+            in (k∈kl₁ , there k,v∈xs)
+    ...   | no k'∉kl₁     | k,v∈l₁xs =
+            let (k∈kl₁ , k,v∈xs) = restrict-needs-both k,v∈l₁xs
+            in (k∈kl₁ , there k,v∈xs)
+
+
 
 Map : Set (a ⊔ b)
 Map = Σ (List (A × B)) (λ l → Unique (keys l))