-- Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
theorem ContMDiffFiberwiseLinear.locality_aux₂
    (n : WithTop ℕ∞) (e : PartialHomeomorph (B × F) (B × F)) (U : Set B) (hU : e.source = U ×ˢ univ)
    (h : ∀ x ∈ U,
      ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_hUu : u ⊆ U) (_hux : x ∈ u)
        (hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : F →L[𝕜] F)) u)
        (h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ((φ x).symm : F →L[𝕜] F)) u),
          (e.restr (u ×ˢ univ)).EqOnSource
            (FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
    ∃ (Φ : B → F ≃L[𝕜] F) (U : Set B) (hU₀ : IsOpen U) (hΦ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (Φ x : F →L[𝕜] F)) U) (h2Φ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ((Φ x).symm : F →L[𝕜] F)) U),
      e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ hΦ.continuousOn h2Φ.continuousOn)

Proof state:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q => (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x => φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y => if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
⊢ ∃ Φ U,
    ∃ (hU₀ : IsOpen U) (hΦ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(Φ x)) U) (h2Φ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(Φ x).symm) U),
      e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ ⋯ ⋯)

Tactic:
have hΦ : ∀ (y) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩ := fun y hy => dif_pos hy

Result:
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q => (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x => φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y => if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
⊢ ∃ Φ U,
    ∃ (hU₀ : IsOpen U) (hΦ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(Φ x)) U) (h2Φ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(Φ x).symm) U),
      e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ ⋯ ⋯)

Proof state:
case h
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q => (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
huφ : ∀ (x x' : ↑U), ∀ y ∈ u x, y ∈ u x' → φ x y = φ x' y
hUu' : U = ⋃ i, u i
hU' : IsOpen U
Φ₀ : ↑U → F ≃L[𝕜] F := iUnionLift u (fun x => φ x ∘ Subtype.val) huφ U ⋯
Φ : B → F ≃L[𝕜] F := fun y => if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩
hΦφ : ∀ (x : ↑U), ∀ y ∈ u x, Φ y = φ x y
x : B
hx : x ∈ U
y : B
hy : y ∈ u ⟨x, hx⟩
⊢ ↑(Φ y) = ↑(φ ⟨x, hx⟩ y)

Tactic:
rw [hΦφ ⟨x, hx⟩ y hy]

Result:
no goals

Proof state:
case h.h
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q => (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
p p' : ↑U
y : B
hyp : y ∈ u p
hyp' : y ∈ u p'
v : F
⊢ (φ p y) v = (φ p' y) v

Tactic:
have h1 : e (y, v) = (y, φ p y v) := heuφ _ ⟨(id hyp : (y, v).fst ∈ u p), trivial⟩

Result:
case h.h
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
U : Set B
hU : e.source = U ×ˢ univ
φ : ↑U → B → F ≃L[𝕜] F
u : ↑U → Set B
hu : ∀ (x : ↑U), IsOpen (u x)
hUu : ∀ (x : ↑U), u x ⊆ U
hux : ∀ (x : ↑U), ↑x ∈ u x
hφ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑U), (e.restr (u x ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
heuφ : ∀ (x : ↑U), EqOn (↑e) (fun q => (q.1, (φ x q.1) q.2)) (u x ×ˢ univ)
p p' : ↑U
y : B
hyp : y ∈ u p
hyp' : y ∈ u p'
v : F
h1 : ↑e (y, v) = (y, (φ p y) v)
⊢ (φ p y) v = (φ p' y) v