{R : Type u_1} →
  [inst : SeminormedCommRing R] →
    {S : Type u_2} →
      [inst_1 : Ring S] →
        [inst_2 : Algebra R S] →
          (toFun : S → ℝ) →
            toFun 0 = 0 →
              (∀ (r s : S), toFun (r + s) ≤ toFun r + toFun s) →
                (∀ (r : S), toFun (-r) = toFun r) →
                  (∀ (x y : S), toFun (x * y) ≤ toFun x * toFun y) →
                    (∀ (x : S), toFun x = 0 → x = 0) →
                      (∀ (a : R) (x : S), toFun (a • x) = ‖a‖ * toFun x) → AlgebraNorm R S