Resolvent sub

lemma resolvent_sub    {T₁ T₂ : H →ₗ.[ℂ] H} (hT : T₂.domain ≤ T₁.domain) {z : ℂ} (hz₁ : z ∈ ρ T₁) (hz₂ : z ∈ ρ T₂) :    𝑅 T₁ z - 𝑅 T₂ z = 𝑅 T₁ z * (T₂ - T₁) * 𝑅 T₂ z := by  symm  calc    _ = 𝑅 T₁ z ∘ᵣ ((T₂ - z • 1 - (T₁ - z • 1)) ∘ᵣ 𝑅 T₂ z) := by      rw [mul_assoc]      congr 2      exact (eq_of_le_of_domain_eq (sub_sub_sub_le_cancel_right _ _ _) (by simp [sub_domain])).symm    _ = 𝑅 T₁ z ∘ᵣ ((T₂ - z • 1) ∘ᵣ 𝑅 T₂ z - (T₁ - z • 1) ∘ᵣ 𝑅 T₂ z) := by      congr      exact sub_compRestricted _ _ _    _ = 𝑅 T₁ z ∘ᵣ (1 - (T₁ - z • 1) ∘ᵣ 𝑅 T₂ z) := by      congr      rw [compRestricted_inverse_eq hz₂.1, inverse_domain, hz₂.2.1]      simp [eq_of_le_of_domain_eq domRestrict_le]    _ = 𝑅 T₁ z - 𝑅 T₁ z ∘ᵣ ((T₁ - z • 1) ∘ᵣ 𝑅 T₂ z) := by      nth_rw 2 [← mul_one (𝑅 T₁ z)]      refine (eq_of_le_of_domain_eq (compRestricted_sub_ge _ _ _) ?_).symm      simp [sub_domain, compRestricted_domain, inverse_domain, hz₁.2]    _ = 𝑅 T₁ z - (domRestrict 1 T₁.domain) ∘ᵣ 𝑅 T₂ z := by      simp [← compRestricted_assoc, inverse_compRestricted_eq hz₁.1, sub_domain]    _ = 𝑅 T₁ z - 𝑅 T₂ z := by      ext x      · suffices 𝑅 T₂ z ⟨x, by simp [inverse_domain, hz₂.2]⟩ ∈ T₁.domain by          simp [sub_domain, mem_compRestricted_domain_iff, inverse_domain, hz₁.2, hz₂.2, this]        have hR₂ : (𝑅 T₂ z).toFun.range = T₂.domain := by simp [inverse_range hz₂.1, sub_domain]        exact hT (hR₂ ▸ mem_range_self _)      · rfl