∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {H' : Type u_5} [inst_6 : TopologicalSpace H']
  {E' : Type u_6} [inst_7 : NormedAddCommGroup E'] [inst_8 : NormedSpace 𝕜 E'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] [IsManifold I 2 M]
  [inst_12 : IsManifold I' 2 M'] [CompleteSpace E] [CompleteSpace E'] {f : M → M'} {V W : (x : M') → TangentSpace I' x}
  {x₀ : M} {s : Set M} {t : Set M'},
  MDifferentiableWithinAt I' I'.tangent (fun x => { proj := x, snd := V x }) t (f x₀) →
    MDifferentiableWithinAt I' I'.tangent (fun x => { proj := x, snd := W x }) t (f x₀) →
      UniqueMDiffOn I s →
        ContMDiffOn I I' 2 f s →
          x₀ ∈ s →
            t ⊆ (extChartAt I' (f x₀)).source →
              Set.MapsTo f s t →
                IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
                    (↑(extChartAt I x₀).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x₀) x₀) →
                  VectorField.mpullbackWithin I I' f (VectorField.mlieBracketWithin I' V W t) s x₀ =
                    VectorField.mlieBracketWithin I (VectorField.mpullbackWithin I I' f V s)
                      (VectorField.mpullbackWithin I I' f W s) s x₀