∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {n : WithTop ℕ∞}
  {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} {s : Set E},
  (∀ x ∈ s, (p x 0).curry0 = f x) →
    (∀ (m : ℕ), ↑m < n → ∀ x ∈ s, HasFDerivWithinAt (fun x => p x m) (p x m.succ).curryLeft s x) →
      (∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => p x m) s) → HasFTaylorSeriesUpToOn n f p s