Prox-regularity of spectral functions and spectral sets

Important properties such as differentiability and convexity of symmetric functions in R(n) can be transferred to the corresponding spectral functions and vice-versa. Continuing to built on this line of research, we hereby prove that a spectral function F: S(n) -> R U {+infinity} is prox-regular if and only if the underlying symmetric function f : R(n) R U {+infinity} is prox-regular. Relevant properties of symmetric sets are also discussed.