  Using the ETAS branching model of triggered seismicity, we apply the
formalism of generating probability functions to calculate exactly the average
difference between the magnitude of a mainshock and the magnitude of its
largest aftershock over all generations. This average magnitude difference is
found empirically to be independent of the mainshock magnitude and equal to
1.2, a universal behavior known as Bath's law. Our theory shows that Bath's law
holds only sufficiently close to the critical regime of the ETAS branching
process. Allowing for error bars +- 0.1 for Bath's constant value around 1.2,
our exact analytical treatment of Bath's law provides new constraints on the
productivity exponent alpha and the branching ratio n: $0.9 <= alpha <= 1$ and
0.8 <= n <= 1. We propose a novel method for measuring alpha based on the
predicted renormalization of the Gutenberg-Richter distribution of the
magnitudes of the largest aftershock. We also introduce the ``second Bath's law
for foreshocks: the probability that a main earthquake turns out to be the
foreshock does not depend on its magnitude.
