  The cross sections of high-energy $\mu^+\mu^-$ pair production and
paradimuonium production by a relativistic electron in the atomic field are
discussed. The calculation is performed exactly in the parameters of the atomic
field. Though the Coulomb corrections to the cross sections, related to the
multiple Coulomb exchange of $\mu^+,\,\mu^-$ and an atom, are negligible, the
Coulomb corrections to the differential cross sections, related to the electron
interaction with an atom, are large. However, the latter Coulomb corrections to
the cross sections integrated over the final electron momentum are small.
Apparently, this effect can easily be observed experimentally. Furthermore, it
is shown that the asymmetry of the cross section with respect to the
permutation of the momenta of $\mu^-$ and $\mu^+$ is large.
