In this chapter we treat the main theme of this work on the efficient use of semiconvex functions in general potential theories. We begin with the Almost Everywhere Theorem which says that in the potential theory defined by a given subequation constraint set \(\mathcal F\) , a locally semiconvex function is \({\mathcal F}\) -subharmonic on an open set if it is \({\mathcal F}\) -subharmonic on a set of full (Lebesgue) measure. Moreover, by Alexandrov’s Theorem and coherence, \({\mathcal F}\) -subharmonicity reduces to a classical statement almost everywhere. Next we discuss the Subharmonic Addition Theorem which gives conditions under which the algebraic sum formula of jet addition implies the functional analytic sum formula of subharmonic addition for the associated subharmonics. Combined with monotonicity and duality this leads to a robust method for establishing comparison principles, which will be discussed in the following chapter.