/-- An identity local homeomorphism belongs to the analytic groupoid. -/
theorem ofSet_mem_analyticGroupoid {s : Set H} (hs : IsOpen s) :
    PartialHomeomorph.ofSet s hs ∈ analyticGroupoid I := by
  rw [analyticGroupoid]
  refine And.intro (ofSet_mem_contDiffGroupoid ∞ I hs) ?_
  

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝ : TopologicalSpace M
s : Set H
hs : IsOpen s
⊢ PartialHomeomorph.ofSet s hs ∈
    (Pregroupoid.groupoid
        {
          property := fun f s =>
            AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ∧
              ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) '' (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ⊆
                interior (range ↑I),
          comp :=
            (_ :
              ∀ {f g : H → H} {u v : Set H},
                AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                      (↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I)) ∧
                    ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) '' (↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I)) ⊆
                      interior (range ↑I) →
                  AnalyticOn 𝕜 (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))
                        (↑(ModelWithCorners.symm I) ⁻¹' v ∩ interior (range ↑I)) ∧
                      ↑I ∘ g ∘ ↑(ModelWithCorners.symm I) '' (↑(ModelWithCorners.symm I) ⁻¹' v ∩ interior (range ↑I)) ⊆
                        interior (range ↑I) →
                    IsOpen u →
                      IsOpen v →
                        IsOpen (u ∩ f ⁻¹' v) →
                          (fun f s =>
                              AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                                  (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ∧
                                ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) ''
                                    (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ⊆
                                  interior (range ↑I))
                            (g ∘ f) (u ∩ f ⁻¹' v)),
          id_mem :=
            (_ :
              AnalyticOn 𝕜 (↑I ∘ id ∘ ↑(ModelWithCorners.symm I))
                  (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ interior (range ↑I)) ∧
                ↑I ∘ id ∘ ↑(ModelWithCorners.symm I) '' (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ interior (range ↑I)) ⊆
                  interior (range ↑I)),
          locality :=
            (_ :
              ∀ {f : H → H} {u : Set H},
                IsOpen u →
                  (∀ x ∈ u,
                      ∃ v,
                        IsOpen v ∧
                          x ∈ v ∧
                            (fun f s =>
                                AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                                    (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ∧
                                  ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) ''
                                      (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ⊆
                                    interior (range ↑I))
                              f (u ∩ v)) →
                    (fun f s =>
                        AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                            (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ∧
                          ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) ''
                              (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ⊆
                            interior (range ↑I))
                      f u),
          congr :=
            (_ :
              ∀ {f g : H → H} {u : Set H},
                IsOpen u →
                  (∀ x ∈ u, g x = f x) →
                    AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                          (↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I)) ∧
                        ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) ''
                            (↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I)) ⊆
                          interior (range ↑I) →
                      (fun f s =>
                          AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                              (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ∧
                            ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) ''
                                (↑(ModelWithCorners.symm I) ⁻¹' s ∩ interior (range ↑I)) ⊆
                              interior (range ↑I))
                        g u) }).members

Next tactic:
apply mem_groupoid_of_pregroupoid.mpr