diff --git a/Lattice.agda b/Lattice.agda
index caa714c..b4097e8 100644
--- a/Lattice.agda
+++ b/Lattice.agda
@@ -112,12 +112,12 @@ module IsEquivalenceInstances where
                 , ⊆-trans m₃ m₂ m₁ m₃⊆m₂ m₂⊆m₁
                 )
 
-            LiftEquivalence : IsEquivalence Map _≈_
-            LiftEquivalence = record
-                { ≈-refl = λ {m₁} → ≈-refl m₁
-                ; ≈-sym = λ {m₁} {m₂} → ≈-sym m₁ m₂
-                ; ≈-trans = λ {m₁} {m₂} {m₃} → ≈-trans m₁ m₂ m₃
-                }
+        LiftEquivalence : IsEquivalence Map _≈_
+        LiftEquivalence = record
+            { ≈-refl = λ {m₁} → ≈-refl m₁
+            ; ≈-sym = λ {m₁} {m₂} → ≈-sym m₁ m₂
+            ; ≈-trans = λ {m₁} {m₂} {m₃} → ≈-trans m₁ m₂ m₃
+            }
 
 module IsSemilatticeInstances where
     module ForNat where
@@ -216,6 +216,34 @@ module IsSemilatticeInstances where
             ; ⊔-idemp = ⊔-idemp
             }
 
+    module ForMap {a} {A B : Set a}
+        (≡-dec-A : Decidable (_≡_ {a} {A}))
+        (_≈₂_ : B → B → Set a)
+        (_⊔₂_ : B → B → B)
+        (sB : IsSemilattice B _≈₂_ _⊔₂_) where
+
+        open import Map A B ≡-dec-A
+
+        private
+            infix 4 _≈_
+            infixl 20 _⊔_
+
+            _≈_ : Map → Map → Set a
+            _≈_ = lift (_≈₂_)
+
+            _⊔_ : Map → Map → Map
+            m₁ ⊔ m₂ = union _⊔₂_ m₁ m₂
+
+        module MapEquiv = IsEquivalenceInstances.ForMap A B ≡-dec-A _≈₂_ (IsSemilattice.≈-equiv sB)
+
+        MapIsUnionSemilattice : IsSemilattice Map _≈_ _⊔_
+        MapIsUnionSemilattice = record
+            { ≈-equiv = MapEquiv.LiftEquivalence
+            ; ⊔-assoc = union-assoc _≈₂_ (IsSemilattice.≈-refl sB) (IsSemilattice.≈-sym sB) _⊔₂_ (IsSemilattice.⊔-assoc sB)
+            ; ⊔-comm = union-comm _≈₂_ (IsSemilattice.≈-refl sB) (IsSemilattice.≈-sym sB) _⊔₂_ (IsSemilattice.⊔-comm sB)
+            ; ⊔-idemp = union-idemp _≈₂_ (IsSemilattice.≈-refl sB) (IsSemilattice.≈-sym sB) _⊔₂_ (IsSemilattice.⊔-idemp sB)
+            }
+
 module IsLatticeInstances where
     module ForNat where
         open Nat
diff --git a/Map.agda b/Map.agda
index 1079cbd..c0bd28e 100644
--- a/Map.agda
+++ b/Map.agda
@@ -342,95 +342,97 @@ module _ (_≈_ : B → B → Set b) where
     lift : Map → Map → Set (a ⊔ b)
     lift m₁ m₂ = subset m₁ m₂ × subset m₂ m₁
 
-module _ (f : B → B → B) where
-    module I = ImplInsert f
+    module _ (≈-refl : ∀ {b : B} → b ≈ b)
+             (≈-sym : ∀ {b₁ b₂ : B} → b₁ ≈ b₂ → b₂ ≈ b₁)
+             (f : B → B → B) where
+        module I = ImplInsert f
 
-    module _ (f-idemp : ∀ (b : B) → f b b ≡ b) where
-        union-idemp : ∀ (m : Map) → lift (_≡_) (union f m m) m
-        union-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
-            where
-                mm-m-subset : subset (_≡_) (union f m m) m
-                mm-m-subset k v k,v∈mm
-                    with Expr-Provenance f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
-                ...   | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
-                        rewrite Map-functional {m = m} v'∈m v''∈m
-                        rewrite Map-functional {m = union f m m} k,v∈mm v'v''∈mm =
-                        (v'' , (f-idemp v'' , v''∈m))
-                ...   | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m))
-                ...   | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m))
+        module _ (f-idemp : ∀ (b : B) → f b b ≈ b) where
+            union-idemp : ∀ (m : Map) → lift (union f m m) m
+            union-idemp m@(l , u) = (mm-m-subset , m-mm-subset)
+                where
+                    mm-m-subset : subset (union f m m) m
+                    mm-m-subset k v k,v∈mm
+                        with Expr-Provenance f k ((` m) ∪ (` m)) (∈-cong proj₁ k,v∈mm)
+                    ...   | (_ , (bothᵘ (single {v'} v'∈m) (single {v''} v''∈m) , v'v''∈mm))
+                            rewrite Map-functional {m = m} v'∈m v''∈m
+                            rewrite Map-functional {m = union f m m} k,v∈mm v'v''∈mm =
+                            (v'' , (f-idemp v'' , v''∈m))
+                    ...   | (_ , (in₁ (single {v'} v'∈m) k∉km , _)) = absurd (k∉km (∈-cong proj₁ v'∈m))
+                    ...   | (_ , (in₂ k∉km (single {v''} v''∈m) , _)) = absurd (k∉km (∈-cong proj₁ v''∈m))
 
-                m-mm-subset : subset (_≡_) m (union f m m)
-                m-mm-subset k v k,v∈m = (f v v , (sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m))
+                    m-mm-subset : subset m (union f m m)
+                    m-mm-subset k v k,v∈m = (f v v , (≈-sym (f-idemp v) , I.union-combines u u k,v∈m k,v∈m))
 
-    module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≡ f b₂ b₁) where
-        union-comm : ∀ (m₁ m₂ : Map) → lift (_≡_) (union f m₁ m₂) (union f m₂ m₁)
-        union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
-            where
-                union-comm-subset : ∀ (m₁ m₂ : Map) → subset (_≡_) (union f m₁ m₂) (union f m₂ m₁)
-                union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
-                    with Expr-Provenance f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
-                ...   | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
-                        rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
-                        (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
-                ...   | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
-                        rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
-                        (v₁ , (refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁))
-                ...   | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
-                        rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
-                        (v₂ , (refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))
+        module _ (f-comm : ∀ (b₁ b₂ : B) → f b₁ b₂ ≈ f b₂ b₁) where
+            union-comm : ∀ (m₁ m₂ : Map) → lift (union f m₁ m₂) (union f m₂ m₁)
+            union-comm m₁ m₂ = (union-comm-subset m₁ m₂ , union-comm-subset m₂ m₁)
+                where
+                    union-comm-subset : ∀ (m₁ m₂ : Map) → subset (union f m₁ m₂) (union f m₂ m₁)
+                    union-comm-subset m₁@(l₁ , u₁) m₂@(l₂ , u₂) k v k,v∈m₁m₂
+                        with Expr-Provenance f k ((` m₁) ∪ (` m₂)) (∈-cong proj₁ k,v∈m₁m₂)
+                    ...   | (_ , (bothᵘ {v₁} {v₂} (single v₁∈m₁) (single v₂∈m₂) , v₁v₂∈m₁m₂))
+                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁v₂∈m₁m₂ =
+                            (f v₂ v₁ , (f-comm v₁ v₂ , I.union-combines u₂ u₁ v₂∈m₂ v₁∈m₁))
+                    ...   | (_ , (in₁ {v₁} (single v₁∈m₁) k∉km₂ , v₁∈m₁m₂))
+                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₁∈m₁m₂ =
+                            (v₁ , (≈-refl , I.union-preserves-∈₂ k∉km₂ v₁∈m₁))
+                    ...   | (_ , (in₂ {v₂} k∉km₁ (single v₂∈m₂) , v₂∈m₁m₂))
+                            rewrite Map-functional {m = union f m₁ m₂} k,v∈m₁m₂ v₂∈m₁m₂ =
+                            (v₂ , (≈-refl , I.union-preserves-∈₁ u₂ v₂∈m₂ k∉km₁))
 
-    module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≡ f b₁ (f b₂ b₃))  where
-        union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
-        union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
-            where
-                union-assoc₁ : subset (_≡_) (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
-                union-assoc₁ k v k,v∈m₁₂m₃
-                    with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
-                ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
-                        let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
-                        in (v₃ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
-                ...   | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃))
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ =
-                        (v₂ , (refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
-                ...   | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃))
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ =
-                        (f v₂ v₃ , (refl , I.union-preserves-∈₂ k∉ke₁  (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
-                ...   | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃))
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ =
-                        (v₁ , (refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃)))
-                ...   | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃))
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ =
-                        (f v₁ v₃ , (refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
-                ...   | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃)
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ =
-                        (f v₁ v₂ , (refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
-                ...   | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
-                        rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
-                        (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
+        module _ (f-assoc : ∀ (b₁ b₂ b₃ : B) → f (f b₁ b₂) b₃ ≈ f b₁ (f b₂ b₃))  where
+            union-assoc : ∀ (m₁ m₂ m₃ : Map) → lift (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
+            union-assoc m₁@(l₁ , u₁) m₂@(l₂ , u₂) m₃@(l₃ , u₃) = (union-assoc₁ , union-assoc₂)
+                where
+                    union-assoc₁ : subset (union f (union f m₁ m₂) m₃) (union f m₁ (union f m₂ m₃))
+                    union-assoc₁ k v k,v∈m₁₂m₃
+                        with Expr-Provenance f k (((` m₁) ∪ (` m₂)) ∪ (` m₃)) (∈-cong proj₁ k,v∈m₁₂m₃)
+                    ...   | (_ , (in₂ k∉ke₁₂ (single {v₃} v₃∈e₃) , v₃∈m₁₂m₃))
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₃∈m₁₂m₃ =
+                            let (k∉ke₁ , k∉ke₂) = I.∉-union-∉-either {l₁ = l₁} {l₂ = l₂} k∉ke₁₂
+                            in (v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
+                    ...   | (_ , (in₁ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) k∉ke₃ , v₂∈m₁₂m₃))
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂∈m₁₂m₃ =
+                            (v₂ , (≈-refl , I.union-preserves-∈₂ k∉ke₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
+                    ...   | (_ , (bothᵘ (in₂ k∉ke₁ (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₂v₃∈m₁₂m₃))
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₂v₃∈m₁₂m₃ =
+                            (f v₂ v₃ , (≈-refl , I.union-preserves-∈₂ k∉ke₁  (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
+                    ...   | (_ , (in₁ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) k∉ke₃ , v₁∈m₁₂m₃))
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁∈m₁₂m₃ =
+                            (v₁ , (≈-refl , I.union-preserves-∈₁ u₁ v₁∈e₁ (I.union-preserves-∉ k∉ke₂ k∉ke₃)))
+                    ...   | (_ , (bothᵘ (in₁ (single {v₁} v₁∈e₁) k∉ke₂) (single {v₃} v₃∈e₃) , v₁v₃∈m₁₂m₃))
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₃∈m₁₂m₃ =
+                            (f v₁ v₃ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₂ k∉ke₂ v₃∈e₃)))
+                    ...   | (_ , (in₁ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) k∉ke₃), v₁v₂∈m₁₂m₃)
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂∈m₁₂m₃ =
+                            (f v₁ v₂ , (≈-refl , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-preserves-∈₁ u₂ v₂∈e₂ k∉ke₃)))
+                    ...   | (_ , (bothᵘ (bothᵘ (single {v₁} v₁∈e₁) (single {v₂} v₂∈e₂)) (single {v₃} v₃∈e₃) , v₁v₂v₃∈m₁₂m₃))
+                            rewrite Map-functional {m = union f (union f m₁ m₂) m₃} k,v∈m₁₂m₃ v₁v₂v₃∈m₁₂m₃ =
+                            (f v₁ (f v₂ v₃) , (f-assoc v₁ v₂ v₃ , I.union-combines u₁ (I.union-preserves-Unique l₂ l₃ u₃) v₁∈e₁ (I.union-combines u₂ u₃ v₂∈e₂ v₃∈e₃)))
 
-                union-assoc₂ : subset (_≡_) (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃) 
-                union-assoc₂ k v k,v∈m₁m₂₃
-                    with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
-                ...   | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
-                        (v₃ , (refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))
-                ...   | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ =
-                        (v₂ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃))
-                ...   | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ =
-                        (f v₂ v₃ , (refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃))
-                ...   | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ =
-                        let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃
-                        in (v₁ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃))
-                ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ =
-                        (f v₁ v₃ , (refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃))
-                ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ =
-                        (f v₁ v₂ , (refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃))
-                ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
-                        rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
-                        (f (f v₁ v₂) v₃ , (sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))
+                    union-assoc₂ : subset (union f m₁ (union f m₂ m₃)) (union f (union f m₁ m₂) m₃) 
+                    union-assoc₂ k v k,v∈m₁m₂₃
+                        with Expr-Provenance f k ((` m₁) ∪ ((` m₂) ∪ (` m₃))) (∈-cong proj₁ k,v∈m₁m₂₃)
+                    ...   | (_ , (in₂ k∉ke₁ (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₃∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₃∈m₁m₂₃ =
+                            (v₃ , (≈-refl , I.union-preserves-∈₂ (I.union-preserves-∉ k∉ke₁ k∉ke₂) v₃∈e₃))
+                    ...   | (_ , (in₂ k∉ke₁ (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₂∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂∈m₁m₂₃ =
+                            (v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) k∉ke₃))
+                    ...   | (_ , (in₂ k∉ke₁ (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₂v₃∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₂v₃∈m₁m₂₃ =
+                            (f v₂ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₂ k∉ke₁ v₂∈e₂) v₃∈e₃))
+                    ...   | (_ , (in₁ (single {v₁} v₁∈e₁) k∉ke₂₃ , v₁∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁∈m₁m₂₃ =
+                            let (k∉ke₂ , k∉ke₃) = I.∉-union-∉-either {l₁ = l₂} {l₂ = l₃} k∉ke₂₃
+                            in (v₁ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) k∉ke₃))
+                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₂ k∉ke₂ (single {v₃} v₃∈e₃)) , v₁v₃∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₃∈m₁m₂₃ =
+                            (f v₁ v₃ , (≈-refl , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-preserves-∈₁ u₁ v₁∈e₁ k∉ke₂) v₃∈e₃))
+                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (in₁ (single {v₂} v₂∈e₂) k∉ke₃) , v₁v₂∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂∈m₁m₂₃ =
+                            (f v₁ v₂ , (≈-refl , I.union-preserves-∈₁ (I.union-preserves-Unique l₁ l₂ u₂) (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) k∉ke₃))
+                    ...   | (_ , (bothᵘ (single {v₁} v₁∈e₁) (bothᵘ (single {v₂} v₂∈e₂) (single {v₃} v₃∈e₃)) , v₁v₂v₃∈m₁m₂₃))
+                            rewrite Map-functional {m = union f m₁ (union f m₂ m₃)} k,v∈m₁m₂₃ v₁v₂v₃∈m₁m₂₃ =
+                            (f (f v₁ v₂) v₃ , (≈-sym (f-assoc v₁ v₂ v₃) , I.union-combines (I.union-preserves-Unique l₁ l₂ u₂) u₃ (I.union-combines u₁ u₂ v₁∈e₁ v₂∈e₂) v₃∈e₃))