∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} {I : BoxIntegral.Box ι}
  (f : BoxIntegral.BoxAdditiveMap ι M I₀),
  ↑I ≤ I₀ → ∀ {π : BoxIntegral.Prepartition I}, π.IsPartition → ∑ J ∈ π.boxes, f J = f I