/-- If a function is `C^n` on a domain with unique derivatives, then its bundled derivative is
`C^m` when `m+1 ≤ n`. In this auxiliary lemma, we prove this fact when the source and target space
are model spaces in models with corners. The general fact is proved in
`ContMDiffOn.contMDiffOn_tangentMapWithin` -/
theorem ContMDiffOn.contMDiffOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
    (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n) (hs : UniqueMDiffOn I s) :
    ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
      (π E (TangentSpace I) ⁻¹' s) := by
  have m_le_n : m ≤ n := (le_add_right le_rfl).trans hmn
  have one_le_n : 1 ≤ n := (le_add_left le_rfl).trans hmn
  have U' : UniqueDiffOn 𝕜 (range I ∩ I.symm ⁻¹' s) := fun y hy ↦ by
    simpa only [UniqueMDiffOn, UniqueMDiffWithinAt, hy.1, inter_comm, mfld_simps]
      using hs (I.symm y) hy.2
  rw [contMDiffOn_iff]
  refine' ⟨hf.continuousOn_tangentMapWithin_aux one_le_n hs, fun p q => _⟩
  suffices h :
    ContDiffOn 𝕜 m
      (((fun p : H' × E' => (I' p.fst, p.snd)) ∘ TotalSpace.toProd H' E') ∘
        tangentMapWithin I I' f s ∘
          (TotalSpace.toProd H E).symm ∘ fun p : E × E => (I.symm p.fst, p.snd))
      ((range I ∩ I.symm ⁻¹' s) ×ˢ univ)
  · -- Porting note: was `simpa [(· ∘ ·)] using h`
    convert h using 1
    · ext1 ⟨x, y⟩
      simp only [mfld_simps]; 

Proof state:
case h.e'_10.h.mk
𝕜 : Type u_1
inst✝²⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace 𝕜 E
H : Type u_3
inst✝²¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝²⁰ : TopologicalSpace M
inst✝¹⁹ : ChartedSpace H M
Is : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝¹⁸ : NormedAddCommGroup E'
inst✝¹⁷ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝¹⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹⁵ : TopologicalSpace M'
inst✝¹⁴ : ChartedSpace H' M'
I's : SmoothManifoldWithCorners I' M'
F : Type u_8
inst✝¹³ : NormedAddCommGroup F
inst✝¹² : NormedSpace 𝕜 F
G : Type u_9
inst✝¹¹ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹⁰ : TopologicalSpace N
inst✝⁹ : ChartedSpace G N
Js : SmoothManifoldWithCorners J N
F' : Type u_11
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace 𝕜 F'
G' : Type u_12
inst✝⁶ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_13
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
J's : SmoothManifoldWithCorners J' N'
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
f✝ f₁ : M → M'
s✝ s₁ t : Set M
x✝ : M
m n : ℕ∞
f : H → H'
s : Set H
hf : ContMDiffOn I I' n f s
hmn : m + 1 ≤ n
hs : UniqueMDiffOn I s
m_le_n : m ≤ n
one_le_n : 1 ≤ n
U' : UniqueDiffOn 𝕜 (range ↑I ∩ ↑(ModelWithCorners.symm I) ⁻¹' s)
p : TangentBundle I H
q : TangentBundle I' H'
h :
  ContDiffOn 𝕜 m
    (((fun p => (↑I' p.1, p.2)) ∘ ⇑(TotalSpace.toProd H' E')) ∘
      tangentMapWithin I I' f s ∘ ⇑(TotalSpace.toProd H E).symm ∘ fun p => (↑(ModelWithCorners.symm I) p.1, p.2))
    ((range ↑I ∩ ↑(ModelWithCorners.symm I) ⁻¹' s) ×ˢ univ)
x y : E
⊢ (↑I'
        ((TotalSpace.toProd H' E')
            (tangentMapWithin I I' f s
              ((TotalSpace.toProd H E).symm
                (↑(PartialEquiv.symm (PartialEquiv.prod I.toPartialEquiv (PartialEquiv.refl E))) (x, y))))).1,
      ((TotalSpace.toProd H' E')
          (tangentMapWithin I I' f s
            ((TotalSpace.toProd H E).symm
              (↑(PartialEquiv.symm (PartialEquiv.prod I.toPartialEquiv (PartialEquiv.refl E))) (x, y))))).2) =
    (↑I' (f (↑(ModelWithCorners.symm I) x)),
      (tangentMapWithin I I' f s ((TotalSpace.toProd H E).symm (↑(ModelWithCorners.symm I) x, y))).snd)

Next tactic:
rfl