∀ {𝕜 : Type u_1} {A : Type u_2} [inst : RCLike 𝕜] [inst_1 : NonUnitalNormedRing A] [inst_2 : StarRing A]
  [inst_3 : NormedSpace 𝕜 A] [inst_4 : IsScalarTower 𝕜 A A] [inst_5 : SMulCommClass 𝕜 A A] [inst_6 : StarModule 𝕜 A]
  {p : A → Prop} {p₁ : Unitization 𝕜 A → Prop} (hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x)
  [inst_7 : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁] [inst_8 : CompleteSpace A]
  [inst_9 : CStarRing A] (a : A) (ha : p a), (Unitization.inrNonUnitalStarAlgHom 𝕜 A).comp (cfcₙHom ha) = cfcₙAux ⋯ a ha