∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {ι : Type u_5} (S : ι → Set α)
  (φ φ_1 : (i : ι) → C(↑(S i), β)) (e_φ : φ = φ_1)
  (hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩)
  (hS : ∀ (x : α), ∃ i, S i ∈ nhds x), ContinuousMap.liftCover S φ hφ hS = ContinuousMap.liftCover S φ_1 ⋯ hS