∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {V W V₁ W₁ : E → E} {s t : Set E} {x : E},
  DifferentiableWithinAt 𝕜 V s x →
    Set.EqOn V₁ V t →
      V₁ x = V x →
        DifferentiableWithinAt 𝕜 W s x →
          Set.EqOn W₁ W t →
            W₁ x = W x →
              UniqueDiffWithinAt 𝕜 t x →
                t ⊆ s → VectorField.lieBracketWithin 𝕜 V₁ W₁ t x = VectorField.lieBracketWithin 𝕜 V W s x