∀ {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] (x : X × Y)
  (motive : (∃ s, IsCompact s ∧ s ∈ nhds x.1) → Prop) (x_1 : ∃ s, IsCompact s ∧ s ∈ nhds x.1),
  (∀ (s₁ : Set X) (hc₁ : IsCompact s₁) (h₁ : s₁ ∈ nhds x.1), motive ⋯) → motive x_1