In this paper a theorem is proved and some of its applications discussed. The theorem gives an answer to the question: Under what conditions does the following relation εα±[Eα(N±1)-E0(N)] hold exactly? In the above relation, the εα′s are the single-particle (sp) energiesdetermined by the corresponding sp potential, E0(N) is the exact ground stateenergy of a closed-shell nucleus N, and the Eα(N±1)′s denote the exact energy eigenvalues of its neighbouring N ±1 nucleus.