theorem HasMFDerivWithinAt.comp (hg : HasMFDerivWithinAt I' I'' g u (f x) g')
    (hf : HasMFDerivWithinAt I I' f s x f') (hst : s ⊆ f ⁻¹' u) :
    HasMFDerivWithinAt I I'' (g ∘ f) s x (g'.comp f') := by
  refine' ⟨ContinuousWithinAt.comp hg.1 hf.1 hst, _⟩
  have A :
    HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g ∘ writtenInExtChartAt I I' x f)
      (ContinuousLinearMap.comp g' f' : E →L[𝕜] E'') ((extChartAt I x).symm ⁻¹' s ∩ range I)
      ((extChartAt I x) x) := by
    have :
      (extChartAt I x).symm ⁻¹' (f ⁻¹' (extChartAt I' (f x)).source) ∈
        𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) x :=
      extChartAt_preimage_mem_nhdsWithin I x
        (hf.1.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _ _))
    unfold HasMFDerivWithinAt at *
    rw [← hasFDerivWithinAt_inter' this, ← extChartAt_preimage_inter_eq] at hf ⊢
    have : writtenInExtChartAt I I' x f ((extChartAt I x) x) = (extChartAt I' (f x)) (f x) := by
      simp only [mfld_simps]
    rw [← this] at hg
    apply HasFDerivWithinAt.comp ((extChartAt I x) x) hg.2 hf.2 _
    intro y hy
    simp only [mfld_simps] at hy
    

Proof state:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f f₀ f₁ : M → M'
x : M
s t : Set M
g : M' → M''
u : Set M'
Is : SmoothManifoldWithCorners I M
I's : SmoothManifoldWithCorners I' M'
I''s : SmoothManifoldWithCorners I'' M''
f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))
hg :
  ContinuousWithinAt g u (f x) ∧
    HasFDerivWithinAt (writtenInExtChartAt I' I'' (f x) g) g'
      (↑(PartialEquiv.symm (extChartAt I' (f x))) ⁻¹' u ∩ range ↑I')
      (writtenInExtChartAt I I' x f (↑(extChartAt I x) x))
hf :
  ContinuousWithinAt f s x ∧
    HasFDerivWithinAt (writtenInExtChartAt I I' x f) f'
      (↑(PartialEquiv.symm (extChartAt I x)) ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source) ∩ range ↑I)
      (↑(extChartAt I x) x)
hst : s ⊆ f ⁻¹' u
this✝ :
  ↑(PartialEquiv.symm (extChartAt I x)) ⁻¹' (f ⁻¹' (extChartAt I' (f x)).source) ∈
    𝓝[↑(PartialEquiv.symm (extChartAt I x)) ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x
this : writtenInExtChartAt I I' x f (↑(extChartAt I x) x) = ↑(extChartAt I' (f x)) (f x)
y : E
hy :
  (↑(PartialHomeomorph.symm (chartAt H x)) (↑(ModelWithCorners.symm I) y) ∈ s ∧
      f (↑(PartialHomeomorph.symm (chartAt H x)) (↑(ModelWithCorners.symm I) y)) ∈
        (chartAt H' (f x)).toPartialEquiv.source) ∧
    y ∈ range ↑I
⊢ writtenInExtChartAt I I' x f y ∈ ↑(PartialEquiv.symm (extChartAt I' (f x))) ⁻¹' u ∩ range ↑I'

Next tactic:
have : f (((chartAt H x).symm : H → M) (I.symm y)) ∈ u := hst hy.1.1