  We give a streamlined proof of a quantitative version of a result from [DG1]
which is crucial for the proof of universality in the bulk [DG1] and also at
the edge [DG2] for orthogonal and symplectic ensembles of random matrices. As a
byproduct, this result gives asymptotic information on a certain ratio of the
beta=1,2,4 partition functions for log gases.
