  An exact result concerning the energy transfers between non-linear waves of
thin elastic plate is derived. Following Kolmogorov's original ideas in
hydrodynamical turbulence, but applied to the F\"oppl-von K\'arm\'an equation
for thin plates, the corresponding K\'arm\'an-Howarth-Monin relation and an
equivalent of the $\frac{4}{5}$-Kolmogorov's law is derived. A third-order
structure function involving increments of the amplitude, velocity and the Airy
stress function of a plate, is proven to be equal to $-\varepsilon\, \ell$,
where $\ell$ is a length scale in the inertial range at which the increments
are evaluated and $\varepsilon$ the energy dissipation rate. Numerical data
confirm this law. In addition, a useful definition of the energy fluxes in
Fourier space is introduced and proven numerically to be flat in the inertial
range. The exact results derived in this Letter are valid for both, weak and
strong wave-turbulence. They could be used as a theoretical benchmark of new
wave-turbulence theories and to develop further analogies with hydrodynamical
turbulence.
