/-- Let `e` be a local homeomorphism of `B × F`.  Suppose that at every point `p` in the source of
`e`, there is some neighbourhood `s` of `p` on which `e` is equal to a bi-smooth fiberwise linear
local homeomorphism.
Then the source of `e` is of the form `U ×ˢ univ`, for some set `U` in `B`, and, 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. -/
theorem SmoothFiberwiseLinear.locality_aux₁ (e : PartialHomeomorph (B × F) (B × F))
    (h : ∀ p ∈ e.source, ∃ s : Set (B × F), IsOpen s ∧ p ∈ s ∧
      ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : 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.restr s).EqOnSource
            (FiberwiseLinear.localHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
    ∃ U : Set B, e.source = U ×ˢ univ ∧ ∀ 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) := by
  rw [SetCoe.forall'] at h
  choose s hs hsp φ u hu hφ h2φ heφ using h
  have hesu : ∀ p : e.source, e.source ∩ s p = u p ×ˢ univ := by
    intro p
    rw [← e.restr_source' (s _) (hs _)]
    exact (heφ p).1
  have hu' : ∀ p : e.source, (p : B × F).fst ∈ u p := by
    intro p
    have : (p : B × F) ∈ e.source ∩ s p := ⟨p.prop, hsp p⟩
    simpa only [hesu, mem_prod, mem_univ, and_true_iff] using this
  have heu : ∀ p : e.source, ∀ q : B × F, q.fst ∈ u p → q ∈ e.source := by
    intro p q hq
    have : q ∈ u p ×ˢ (univ : Set F) := ⟨hq, trivial⟩
    rw [← hesu p] at this
    exact this.1
  have he : e.source = (Prod.fst '' e.source) ×ˢ (univ : Set F) := by
    apply HasSubset.Subset.antisymm
    · intro p hp
      exact ⟨⟨p, hp, rfl⟩, trivial⟩
    · rintro ⟨x, v⟩ ⟨⟨p, hp, rfl : p.fst = x⟩, -⟩
      exact heu ⟨p, hp⟩ (p.fst, v) (hu' ⟨p, hp⟩)
  

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)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen (u x)
hφ : ∀ (x : ↑e.source), SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x_1 => ↑(ContinuousLinearEquiv.symm (φ x x_1))) (u x)
heφ :
  ∀ (x : ↑e.source),
    PartialHomeomorph.EqOnSource (PartialHomeomorph.restr e (s x))
      (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)))
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source
he : e.source = (Prod.fst '' e.source) ×ˢ univ
⊢ ∃ U,
    e.source = U ×ˢ univ ∧
      ∀ x ∈ U,
        ∃ φ u,
          ∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ :
            SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) u),
            PartialHomeomorph.EqOnSource (PartialHomeomorph.restr e (u ×ˢ univ))
              (FiberwiseLinear.localHomeomorph φ hu (_ : ContinuousOn (fun x => ↑(φ x)) u)
                (_ : ContinuousOn (fun x => ↑(ContinuousLinearEquiv.symm (φ x))) u))

Next tactic:
refine' ⟨Prod.fst '' e.source, he, _⟩