  Dyson's integration theorem is widely used in the computation of eigenvalue
correlation functions in Random Matrix Theory. Here we focus on the variant of
the theorem for determinants, relevant for the unitary ensembles with Dyson
index beta = 2. We derive a formula reducing the (n-k)-fold integral of an n x
n determinant of a kernel of two sets of arbitrary functions to a determinant
of size k x k. Our generalisation allows for sets of functions that are not
orthogonal or bi-orthogonal with respect to the integration measure. In the
special case of orthogonal functions Dyson's theorem is recovered.
