/-- Let `e` be a local homeomorphism of `B × F` whose source is `U ×ˢ univ`, for some set `U` in
`B`, and which, at any point `x` in `U`, admits a neighbourhood `u` of `x` such that `e` is equal on
`u ×ˢ univ` to some bi-smooth fiberwise linear local homeomorphism.  Then `e` itself is equal to
some bi-smooth fiberwise linear local homeomorphism.

This is the key mathematical point of the `locality` condition in the construction of the
`StructureGroupoid` of bi-smooth fiberwise linear local homeomorphisms.  The proof is by gluing
together the various bi-smooth fiberwise linear local homeomorphism which exist locally.

The `U` in the conclusion is the same `U` as in the hypothesis. We state it like this, because this
is exactly what we need for `smoothFiberwiseLinear`. -/
theorem SmoothFiberwiseLinear.locality_aux₂ (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 : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x : F →L[𝕜] F)) u)
        (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ((φ x).symm : F →L[𝕜] F)) u),
          (e.restr (u ×ˢ univ)).EqOnSource
            (FiberwiseLinear.localHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
    ∃ (Φ : B → F ≃L[𝕜] F) (U : Set B) (hU₀ : IsOpen U) (hΦ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (Φ x : F →L[𝕜] F)) U) (h2Φ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ((Φ x).symm : F →L[𝕜] F)) U),
      e.EqOnSource (FiberwiseLinear.localHomeomorph Φ hU₀ hΦ.continuousOn h2Φ.continuousOn) := by
  classical
  rw [SetCoe.forall'] at h
  choose! φ u hu hUu hux hφ h2φ heφ using h
  have heuφ : ∀ x : U, EqOn e (fun q => (q.1, φ x q.1 q.2)) (u x ×ˢ univ) := fun x p hp ↦ by
    refine' (heφ x).2 _
    rw [(heφ x).1]
    exact hp
  have huφ : ∀ (x x' : U) (y : B), y ∈ u x → y ∈ u x' → φ x y = φ x' y := fun p p' y hyp hyp' ↦ by
    ext v
    have h1 : e (y, v) = (y, φ p y v) := heuφ _ ⟨(id hyp : (y, v).fst ∈ u p), trivial⟩
    have h2 : e (y, v) = (y, φ p' y v) := heuφ _ ⟨(id hyp' : (y, v).fst ∈ u p'), trivial⟩
    exact congr_arg Prod.snd (h1.symm.trans h2)
  have hUu' : U = ⋃ i, u i := by
    ext x
    rw [mem_iUnion]
    refine' ⟨fun h => ⟨⟨x, h⟩, hux _⟩, _⟩
    rintro ⟨x, hx⟩
    exact hUu x hx
  have hU' : IsOpen U := by
    rw [hUu']
    apply isOpen_iUnion hu
  let Φ₀ : U → F ≃L[𝕜] F := iUnionLift u (fun x => φ x ∘ (↑)) huφ U hUu'.le
  let Φ : B → F ≃L[𝕜] F := fun y =>
    if hy : y ∈ U then Φ₀ ⟨y, hy⟩ else ContinuousLinearEquiv.refl 𝕜 F
  have hΦ : ∀ (y) (hy : y ∈ U), Φ y = Φ₀ ⟨y, hy⟩ := fun y hy => dif_pos hy
  have hΦφ : ∀ x : U, ∀ y ∈ u x, Φ y = φ x y := by
    intro x y hyu
    refine' (hΦ y (hUu x hyu)).trans _
    

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
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), SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑U), SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x_1 => ↑(ContinuousLinearEquiv.symm (φ x x_1))) (u x)
heφ :
  ∀ (x : ↑U),
    PartialHomeomorph.EqOnSource (PartialHomeomorph.restr e (u x ×ˢ univ))
      (FiberwiseLinear.localHomeomorph (φ x) (_ : IsOpen (u x)) (_ : ContinuousOn (fun x_1 => ↑(φ x x_1)) (u x))
        (_ : ContinuousOn (fun x_1 => ↑(ContinuousLinearEquiv.symm (φ x x_1))) (u 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 (_ : U ≤ ⋃ i, u i)
Φ : B → F ≃L[𝕜] F := fun y => if hy : y ∈ U then Φ₀ { val := y, property := hy } else ContinuousLinearEquiv.refl 𝕜 F
hΦ : ∀ (y : B) (hy : y ∈ U), Φ y = Φ₀ { val := y, property := hy }
x : ↑U
y : B
hyu : y ∈ u x
⊢ Φ₀ { val := y, property := (_ : y ∈ U) } = φ x y

Next tactic:
exact iUnionLift_mk ⟨y, hyu⟩ _