  It is shown that the controversy on phonons or electrons being the most
influenced heat carriers below the critical temperature of high-T$_c$
superconductors can be resolved. Electrical and thermal properties of the same
DyBa$_2$Cu$_3$O$_{7-x}$ monodomain have been measured for two highly different
oxygenation levels. While the oxygenated sample DyBa$_2$Cu$_3$O$_{7}$ has very
good superconducting properties ($T_c=90$ K), the DyBa$_2$Cu$_3$O$_{6.3}$
sample exhibits an insulator behavior. A careful comparison between
measurements of the {\bf thermal diffusivity} of both samples allows us to
extract the electronic contribution. This contribution to the relaxation time
of heat carriers is shown to be large below $T_c$ and more sensitive to the
superconducting state than the phonon contribution.
