On uniqueness of differential structures on orbifolds

It is known that every C r -orbifold, 1 ≤ r ≤ ∞ , has a compatible C s -differential structure, for every s, where r < s ≤ ω . We prove that if two reduced C r -orbifolds, 2 ≤ r ≤ ω , are C 2 -diffeomorphic, then they are C r -diffeomorphic. It follows that the compatible C s -differential structure on a reduced C r -orbifold, 2 ≤ r < s ≤ ω , is unique up to a C s -diffeomorphism.