  We prove that in random matrix theory there exists a universal relation
between the one-point Green's function $G$ and the connected two- point Green's
function $G_c$ given by \vfill $ N^2 G_c(z,w) = {\part^2 \over \part z \part w}
\log (({G(z)- G(w) \over z -w}) + {\rm {irrelevant \ factorized \ terms.}} $
This relation is universal in the sense that it does not depend on the
probability distribution of the random matrices for a broad class of
distributions, even though $G$ is known to depend on the probability
distribution in detail. The universality discussed here represents a different
statement than the universality we discovered a couple of years ago, which
states that $a^2 G_c(az, aw)$ is independent of the probability distribution,
where $a$ denotes the width of the spectrum and depends sensitively on the
probability distribution. It is shown that the universality proved here also
holds for the more general problem of a Hamiltonian consisting of the sum of a
deterministic term and a random term analyzed perturbatively by Br\'ezin,
Hikami, and Zee.
