/-- Continuous constant functions as a `RingHom`. -/
def ContinuousMap.C : R →+* C(α, A) where
  toFun := fun c : R => ⟨fun _ : α => (algebraMap R A) c, continuous_const⟩
  map_one' := by ext _; exact (algebraMap R A).map_one
  map_mul' c₁ c₂ := by ext _; exact (algebraMap R A).map_mul _ _
  map_zero' := by ext _; exact (algebraMap R A).map_zero
  map_add' c₁ c₂ := by 

Proof state:
α : Type u_1
inst✝⁹ : TopologicalSpace α
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : TopologicalSemiring A
A₂ : Type u_4
inst✝³ : TopologicalSpace A₂
inst✝² : Semiring A₂
inst✝¹ : Algebra R A₂
inst✝ : TopologicalSemiring A₂
c₁ c₂ : R
⊢ OneHom.toFun
      (↑{
          toOneHom :=
            { toFun := fun c => mk fun x => (algebraMap R A) c,
              map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) },
          map_mul' :=
            (_ :
              ∀ (c₁ c₂ : R),
                OneHom.toFun
                    { toFun := fun c => mk fun x => (algebraMap R A) c,
                      map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                    (c₁ * c₂) =
                  OneHom.toFun
                      { toFun := fun c => mk fun x => (algebraMap R A) c,
                        map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                      c₁ *
                    OneHom.toFun
                      { toFun := fun c => mk fun x => (algebraMap R A) c,
                        map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                      c₂) })
      (c₁ + c₂) =
    OneHom.toFun
        (↑{
            toOneHom :=
              { toFun := fun c => mk fun x => (algebraMap R A) c,
                map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) },
            map_mul' :=
              (_ :
                ∀ (c₁ c₂ : R),
                  OneHom.toFun
                      { toFun := fun c => mk fun x => (algebraMap R A) c,
                        map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                      (c₁ * c₂) =
                    OneHom.toFun
                        { toFun := fun c => mk fun x => (algebraMap R A) c,
                          map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                        c₁ *
                      OneHom.toFun
                        { toFun := fun c => mk fun x => (algebraMap R A) c,
                          map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                        c₂) })
        c₁ +
      OneHom.toFun
        (↑{
            toOneHom :=
              { toFun := fun c => mk fun x => (algebraMap R A) c,
                map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) },
            map_mul' :=
              (_ :
                ∀ (c₁ c₂ : R),
                  OneHom.toFun
                      { toFun := fun c => mk fun x => (algebraMap R A) c,
                        map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                      (c₁ * c₂) =
                    OneHom.toFun
                        { toFun := fun c => mk fun x => (algebraMap R A) c,
                          map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                        c₁ *
                      OneHom.toFun
                        { toFun := fun c => mk fun x => (algebraMap R A) c,
                          map_one' := (_ : (fun c => mk fun x => (algebraMap R A) c) 1 = 1) }
                        c₂) })
        c₂

Next tactic:
ext _