∀ {ι : Type u_2} {E : Type u_3} {F : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] {I J : BoxIntegral.Box ι} [inst_4 : Fintype ι]
  {l : BoxIntegral.IntegrationParams} {f : (ι → ℝ) → E} {vol : BoxIntegral.BoxAdditiveMap ι (E →L[ℝ] F) ⊤} (hJ : J ≤ I)
  (motive :
    (∃ y,
        BoxIntegral.HasIntegral J l f vol y ∧
          Filter.Tendsto (BoxIntegral.integralSum f vol)
            (BoxIntegral.IntegrationParams.toFilteriUnion I (BoxIntegral.Prepartition.single I J hJ)) (nhds y)) →
      Prop)
  (x :
    ∃ y,
      BoxIntegral.HasIntegral J l f vol y ∧
        Filter.Tendsto (BoxIntegral.integralSum f vol)
          (BoxIntegral.IntegrationParams.toFilteriUnion I (BoxIntegral.Prepartition.single I J hJ)) (nhds y)),
  (∀ (_y : F) (h₁ : BoxIntegral.HasIntegral J l f vol _y)
      (h₂ :
        Filter.Tendsto (BoxIntegral.integralSum f vol)
          (BoxIntegral.IntegrationParams.toFilteriUnion I (BoxIntegral.Prepartition.single I J hJ)) (nhds _y)),
      motive ⋯) →
    motive x