∀ {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y]
  [inst_2 : TopologicalSpace Z] {g : Y → Z} {f : X → Y} {s : Set X},
  IsLocalHomeomorphOn (g ∘ f) s → IsLocalHomeomorphOn g (f '' s) → (∀ x ∈ s, ContinuousAt f x) → IsLocalHomeomorphOn f s