∀ {ι : Type u_1} [inst : Fintype ι] {I : BoxIntegral.Box ι} {c : NNReal} {l : BoxIntegral.IntegrationParams}
  {r : (ι → ℝ) → ↑(Set.Ioi 0)} {π : BoxIntegral.Prepartition I}
  {πi : (J : BoxIntegral.Box ι) → BoxIntegral.TaggedPrepartition J},
  (∀ J ∈ π, l.MemBaseSet J c r (πi J)) →
    (∀ J ∈ π, (πi J).IsPartition) →
      (l.bDistortion = true → π.compl.distortion ≤ c) → l.MemBaseSet I c r (π.biUnionTagged πi)