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Let $H=\langle n_1, \ldots ,n_4\rangle$ be a numerical semigroup generated by$4$ elements, which is symmetric and let $k[H]$ be the semigroup ring of $H$over a field $k$. H. Bresinski proved that the defining ideal of $k[H]$ isminimally generated by $3$ or $5$ elements. We give a new short proof ofBresinski's Theorem using the structure theorem of Buchsbaum and Eisenbud onthe minimal free resolution of Gorenstein rings of embedding codimension $3$.

A short proof of Bresinski's Theorem on Gorenstein semigroup rings generated by 4 elements