/-- The composition of `C^n` functions within domains at points is `C^n`. -/
theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
    (hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
    (st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g ∘ f) s x := by
  rw [contMDiffWithinAt_iff] at hg hf ⊢
  refine' ⟨hg.1.comp hf.1 st, _⟩
  set e := extChartAt I x
  set e' := extChartAt I' (f x)
  have : e' (f x) = (writtenInExtChartAt I I' x f) (e x) := by simp only [mfld_simps]
  rw [this] at hg
  have A : ∀ᶠ y in 𝓝[e.symm ⁻¹' s ∩ range I] e x, f (e.symm y) ∈ t ∧ f (e.symm y) ∈ e'.source := by
    simp only [← map_extChartAt_nhdsWithin, eventually_map]
    filter_upwards [hf.1.tendsto (extChartAt_source_mem_nhds I' (f x)),
      inter_mem_nhdsWithin s (extChartAt_source_mem_nhds I x)]
    rintro x' (hfx' : f x' ∈ e'.source) ⟨hx's, hx'⟩
    simp only [e.map_source hx', true_and_iff, e.left_inv hx', st hx's, *]
  refine' ((hg.2.comp _ (hf.2.mono (inter_subset_right _ _)) (inter_subset_left _ _)).mono_of_mem
    (inter_mem _ self_mem_nhdsWithin)).congr_of_eventuallyEq _ _
  · filter_upwards [A]
    rintro x' ⟨ht, hfx'⟩
    simp only [*, mem_preimage, writtenInExtChartAt, (· ∘ ·), mem_inter_iff, e'.left_inv,
      true_and_iff]
    exact mem_range_self _
  · filter_upwards [A]
    rintro x' ⟨-, hfx'⟩
    

Proof state:
case h.intro
𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e✝ : PartialHomeomorph M H
e'✝ : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t✝ : Set M
x✝ : M
m n : ℕ∞
t : Set M'
g : M' → M''
x : M
st : MapsTo f s t
e : PartialEquiv M E := extChartAt I x
e' : PartialEquiv M' E' := extChartAt I' (f x)
hg :
  ContinuousWithinAt g t (f x) ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I'' (g (f x))) ∘ g ∘ ↑(PartialEquiv.symm e'))
      (↑(PartialEquiv.symm e') ⁻¹' t ∩ range ↑I') (writtenInExtChartAt I I' x f (↑e x))
hf :
  ContinuousWithinAt f s x ∧
    ContDiffWithinAt 𝕜 n (↑e' ∘ f ∘ ↑(PartialEquiv.symm e)) (↑(PartialEquiv.symm e) ⁻¹' s ∩ range ↑I) (↑e x)
this : ↑e' (f x) = writtenInExtChartAt I I' x f (↑e x)
A :
  ∀ᶠ (y : E) in 𝓝[↑(PartialEquiv.symm e) ⁻¹' s ∩ range ↑I] ↑e x,
    f (↑(PartialEquiv.symm e) y) ∈ t ∧ f (↑(PartialEquiv.symm e) y) ∈ e'.source
x' : E
hfx' : f (↑(PartialEquiv.symm e) x') ∈ e'.source
⊢ (↑(extChartAt I'' ((g ∘ f) x)) ∘ (g ∘ f) ∘ ↑(PartialEquiv.symm e)) x' =
    ((↑(extChartAt I'' (g (f x))) ∘ g ∘ ↑(PartialEquiv.symm e')) ∘ writtenInExtChartAt I I' x f) x'

Next tactic:
simp only [*, (· ∘ ·), writtenInExtChartAt, e'.left_inv]