Is Closable defect Number const

lemma IsClosable.defectNumber_const [CompleteSpace H]    {T : H →ₗ.[ℂ] H} (hT : T.IsClosable)    {z₁ z₂ : ℂ} (hz : z₂ ∈ connectedComponentIn T.regularityDomain z₁) :    T.defectNumber z₁ = T.defectNumber z₂ := by  by_cases hz₁ : z₁ ∈ T.regularityDomain  · have h_joined : JoinedIn T.regularityDomain z₁ z₂ := by      haveI := T.regularityDomain_isOpen.locallyPathConnectedSpace      have hz₂ : z₂ ∈ T.regularityDomain := connectedComponentIn_subset _ _ hz      apply (joinedIn_iff_joined hz₁ hz₂).mpr      rw [← mem_pathComponent_iff, pathComponent_eq_connectedComponent]      exact mem_of_mem_image_val (connectedComponentIn_eq_image hz₁ ▸ hz)    let path : Path z₁ z₂ := h_joined.somePath    by_contra! hne    let a : unitInterval := sSup {r | ∀ r' ≤ r, T.defectNumber (path r') = T.defectNumber z₁}    have ha : ∀ r < a, T.defectNumber (path r) = T.defectNumber z₁ := by      intro r hr      obtain ⟨b, hb, hrb⟩ := lt_sSup_iff.mp hr      exact hb r hrb.le    let c : ℝ := (h_joined.somePath_mem a).choose    have hc_pos : 0 < c := (h_joined.somePath_mem a).choose_spec.1    have hc_bound : IsLowerBound T (path a) c := (h_joined.somePath_mem a).choose_spec.2    obtain ⟨ε, hε, hε_ball⟩ : ∃ ε > 0, ball a ε ⊆ path ⁻¹' ball (path a) c := by      apply Metric.mem_nhds_iff.mp      refine (IsOpen.mem_nhds_iff ?_).mpr ?_      · exact path.continuous.isOpen_preimage _ isOpen_ball      · simp [hc_pos]    obtain ⟨b₁, h₁, h₁'⟩ : ∃ b ∈ ball a ε, T.defectNumber (path b) = T.defectNumber z₁ := by      rcases le_or_gt ε a with hle | hlt      · let r : ℝ := a - ε / 2        have hr : 0 ≤ r := by dsimp [r]; linarith        have hr' : r < a := sub_lt_self _ (half_pos hε)        use ⟨r, hr, by linarith [a.2.2]⟩        exact ⟨by simp [dist, r, abs_div, abs_of_nonneg hε.le, hε], ha _ hr'⟩      · exact ⟨0, by simp [dist, abs_of_nonneg a.2.1, hlt], by rw [path.source]⟩    obtain ⟨b₂, h₂, h₂'⟩ : ∃ b ∈ ball a ε, T.defectNumber (path b) ≠ T.defectNumber z₁ := by      by_cases! h₀ : a < 1      · by_contra! h'        let r : unitInterval :=          ⟨min (a + ε / 2) 1, le_inf_iff.mpr ⟨by linarith [a.2.1], zero_le_one⟩, inf_le_right⟩        refine not_le_of_gt (a := a) (b := r) ?_ ?_        · apply (Set.inclusion_lt_inclusion <| Set.subset_univ _).mp          simp [r, hε, h₀]        · refine le_sSup_iff.mpr fun _ hub ↦ hub fun b hbr ↦ ?_          rcases lt_or_ge b a with hlt | hle          · exact ha b hlt          · refine h' b ?_            apply mem_ball.mpr            calc              _ = (b : ℝ) - a := by simp [dist, hle]              _ ≤ r - a := by simp [hbr]              _ = min (ε / 2) (1 - a) := by simp [r, ← min_sub_sub_right]              _ < ε := by simp [hε]      · have : a = 1 := eq_of_le_of_ge a.2.2 h₀        refine ⟨a, mem_ball_self hε, by rw [this, path.target]; exact hne.symm⟩    apply h₁' ▸ h₂'    rw [← defectNumber_eq_of_mem_ball hT hc_bound (hε_ball h₁)]    rw [← defectNumber_eq_of_mem_ball hT hc_bound (hε_ball h₂)]  · exact ((mem_empty_iff_false z₂).mp (connectedComponentIn_eq_empty hz₁ ▸ hz)).elim