∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {V W V₁ : E → E} {s : Set E} {x : E},
  DifferentiableWithinAt 𝕜 V s x →
    DifferentiableWithinAt 𝕜 V₁ s x →
      UniqueDiffWithinAt 𝕜 s x →
        VectorField.lieBracketWithin 𝕜 (V + V₁) W s x =
          VectorField.lieBracketWithin 𝕜 V W s x + VectorField.lieBracketWithin 𝕜 V₁ W s x