  By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove
that the class of functions ${\mit \Xi}(z)$ with an integral representation of
the form $\int_{0}^{+\infty}du\,{\mit \Omega}(u)\,{\rm ch}(uz)$ with a
real-valued function ${\mit \Omega}(u) \ge 0$ which is non-increasing and
decreases in infinity more rapidly than any exponential functions
$\exp\left(-\lambda u\right),\,\lambda >0$ possesses zeros only on the
imaginary axis. The Riemann zeta function $\zeta(s)$ as it is known can be
related to an entire function $\xi(s)$ with the same non-trivial zeros as
$\zeta(s)$. Then after a trivial argument displacement $s\leftrightarrow
z=s-\frac{1}{2}$ we relate it to a function ${\mit \Xi}(z)$ with a
representation of the form ${\mit \Xi}(z)=\int_{0}^{+\infty}du\,{\mit
\Omega}(u)\,{\rm ch}(uz)$ where ${\mit \Omega}(u)$ is rapidly decreasing in
infinity and satisfies all requirements necessary for the given proof of the
position of its zeros on the imaginary axis $z={\rm i} y$ by the second
mean-value theorem. Besides this theorem we apply the Cauchy-Riemann
differential equation in an integrated operator form derived in the Appendix B.
All this means that we prove a theorem for zeros of ${\mit \Xi}(z)$ on the
imaginary axis $z={\rm i} y$ for a whole class of function ${\mit \Omega}(u)$
which includes in this way the proof of the Riemann hypothesis. This whole
class includes, in particular, the modified Bessel functions ${\rm I}_{\nu}(z)$
for which it is known that their zeros lie on the imaginary axis and which
affirms our conclusions. A class of almost-periodic functions to piece-wise
constant nonincreasing functions ${\rm \Omega}(u)$ belongs also to this case.
At the end we give shortly an equivalent way of a more formal description of
the obtained results using the Mellin transform of functions with its variable
substituted by an operator.
