∀ {X : Type u_3} {Y : Type u_1} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (𝕜 : Type u_4)
  [inst_2 : CommSemiring 𝕜] (A : Type u_2) [inst_3 : TopologicalSpace A] [inst_4 : Semiring A]
  [inst_5 : IsTopologicalSemiring A] [inst_6 : StarRing A] [inst_7 : ContinuousStar A] [inst_8 : Algebra 𝕜 A]
  (f : X ≃ₜ Y) (x y : C(Y, A)),
  (↑↑(ContinuousMap.compStarAlgHom' 𝕜 A ↑f).toRingHom).toFun (x + y) =
    (↑↑(ContinuousMap.compStarAlgHom' 𝕜 A ↑f).toRingHom).toFun x +
      (↑↑(ContinuousMap.compStarAlgHom' 𝕜 A ↑f).toRingHom).toFun y