∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : PseudoEMetricSpace γ] {𝔖 : Set (Set α)}
  [inst_1 : PseudoEMetricSpace β] {f : γ → UniformOnFun α β 𝔖} (K : Set α → NNReal),
  (∀ s ∈ 𝔖, ∀ c ∈ s, LipschitzWith (K s) fun x => (UniformOnFun.toFun 𝔖) (f x) c) → Continuous f