  This article proves the Riemann hypothesis, which states that all non-trivial
zeros of the zeta function have a real part equal to 1/2. We inspect in detail
the integral form of the (symmetrized) completed zeta function, which is a
product between the zeta and gamma functions. It is known that two integral
lines, expressing the completed zeta function, rotated from the real axis in
the opposite directions, can be shifted without affecting the completed zeta
function owing to the residue theorem. The completed zeta function is regular
in the region of the complex plane under consideration. For convenience in the
subsequent singularity analysis of the above integral, we first deform and
shift the integral contours. We then investigate the singularities of the
composite elements (caused by polynomial integrals in opposite directions),
which appear only in the case for which the distance between the contours and
the origin of the coordinates approaches zero. The real part of the zeros of
the zeta function is determined to be 1/2 along a symmetry line from the
singularity removal condition. (In the other points, the singularities are
adequately cancelled as a whole to lead to a finite value.)
