  The estimation of rare K decay matrix-elements from Kl3 experimental data is
extended beyond LO in Chiral Perturbation Theory. Isospin-breaking effects at
NLO (and partially NNLO) in the ChPT expansion, as well as QED radiative
corrections are now accounted for. The analysis relies mainly on the cleanness
of two specific ratios of form-factors, for which the theoretical control is
excellent. As a result, the uncertainties on the K+ --> pi+ nu nubar and KL -->
pi0 nu nubar matrix-elements are reduced by a factor of about 7 and 4,
respectively, and similarly for the direct CP-violating contribution to KL -->
pi0 l+ l-. They could be reduced even further with better experimental data for
the Kl3 slopes and the K+l3 branching ratios. As a result, the non-parametric
errors for B(K --> pi nu nubar) and for the direct CP-violating contributions
to B(KL --> pi0 l+ l-) are now completely dominated by those on the
short-distance physics.
