  Given a quadratic function $h$ that satisfies a Slater condition,
Yakubovich's S-Procedure (or S-Lemma) gives a characterization of all other
quadratic functions that are copositive with $h$ in a form that is amenable to
numerical computations. In this paper we present a deep-rooted connection
between the S-Procedure and the dual cone calculus formula $(K_1\cap K_2)^*=
K_1^*+K_2^*$, which holds for closed convex cones in $\R^2$. To establish the
link with the S-Procedure, we generalize the dual cone calculus formula to a
situation where $K_1$ is nonclosed, nonconvex and nonconic but exhibits
sufficient mathematical resemblance to a closed convex cone. As a result, we
obtain a new proof of the S-Lemma and an extension to Hilbert space kernels.
