We present a microscopic theory of the fractional quantum Hall effect at the filling factor nu with even denominator, which is recently observed in a double-layer electron system. In our approach electrons belonging to different layers are interpreted as different types of anyons with appropriate statistics. The wavefunction of the Hall state is calculated, which is found to coincide with that of Halperin. We also analyze vortex (quasihole) excitations. It is shown that a single vortex carries electric charges on both of the layers; for instance, a vortex at nu = 1/2 has the electric charge 3/8e on one layer and -1/8e on the other layer. The ground state at the vicinity of nu = 1/2 is given by a Wigner crystal made of these vortices.