∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : ℕ} {s : Set E} {x : E}
  {ω : E → E [⋀^Fin (n + 1)]→L[𝕜] F} {V : Fin (n + 2) → E → E},
  DifferentiableWithinAt 𝕜 ω s x →
    (∀ (i : Fin (n + 2)), DifferentiableWithinAt 𝕜 (V i) s x) →
      UniqueDiffWithinAt 𝕜 s x →
        ((extDerivWithin ω s x) fun x_1 => V x_1 x) =
          ∑ i, (-1) ^ ↑i • (fderivWithin 𝕜 (fun x => (ω x) (i.removeNth fun x_1 => V x_1 x)) s x) (V i x) -
            ∑ i,
              ∑ j ∈ Finset.Ici i,
                (-1) ^ (↑i + ↑j) •
                  (ω x)
                    (Matrix.vecCons (VectorField.lieBracketWithin 𝕜 (V i.castSucc) (V j.succ) s x)
                      (j.removeNth (i.castSucc.removeNth fun x_1 => V x_1 x)))