∀ {α : Type u_1} [inst : UniformSpace α] {K : Set α} {U : (x : α) → x ∈ K → Set α} (x : Set (α × α))
  (motive : (x ∈ uniformity α ∧ ∀ x_1 ∈ K, ∃ y, UniformSpace.ball x_1 x ⊆ Kᶜ ∪ U ↑y ⋯) → Prop)
  (x_1 : x ∈ uniformity α ∧ ∀ x_1 ∈ K, ∃ y, UniformSpace.ball x_1 x ⊆ Kᶜ ∪ U ↑y ⋯),
  (∀ (V_uni : x ∈ uniformity α) (hV : ∀ x_2 ∈ K, ∃ y, UniformSpace.ball x_2 x ⊆ Kᶜ ∪ U ↑y ⋯), motive ⋯) → motive x_1