First note that acts properly discontinuously (The deck transformation group acts properly discontinuously) and is a regular covering (Orbit space of a properly discontinuous effective group action). Since is clearly constant for each fibre of , there exists a function such that , and by Universal property this is continuous. Since is surjective so is , and since is regular and thus is transitive is injective, because if it follows

and thus there exists with , implying . Since both and are local homeomorphisms, so is , in particular it is open. Therefore is a homeomoprhism.