Prove a property of multi-key lookup

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Lattice/Map.agda

@@ -1112,6 +1112,19 @@ _[_] m (k ∷ ks)
 ...   | yes k∈km = proj₁ (locate {m = m} k∈km) ∷ (m [ ks ])
 ...   | no _ = m [ ks ]
 
+[]-∈ : ∀ {k : A} {v : B} {ks : List A} (m : Map) →
+       (k , v) ∈ m → k ∈ˡ ks → v ∈ˡ (m [ ks ])
+[]-∈ {k} {v} {ks} m k,v∈m (here refl)
+    with ∈k-dec k (proj₁ m)
+...    | no k∉km = ⊥-elim (k∉km (forget k,v∈m))
+...    | yes k∈km
+         with (v' , k,v'∈m) ← locate {m = m} k∈km
+         rewrite Map-functional {m = m} k,v'∈m k,v∈m = here refl
+[]-∈ {k} {v} {k' ∷ ks'} m k,v∈m (there k∈ks')
+    with ∈k-dec k' (proj₁ m)
+...   | no _  = []-∈ m k,v∈m k∈ks'
+...   | yes _ = there ([]-∈ m k,v∈m k∈ks')
+
 m₁≼m₂⇒m₁[k]≼m₂[k] : ∀ (m₁ m₂ : Map) {k : A} {v₁ v₂ : B} →
                     m₁ ≼ m₂ → (k , v₁) ∈ m₁ → (k , v₂) ∈ m₂ → v₁ ≼₂ v₂
 m₁≼m₂⇒m₁[k]≼m₂[k] m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂