Laser light reflection by a relativistically moving electron density modulation (flying mirror) in a wake wave generated in a plasma by a high intensity laser pulse is investigated experimentally. A counterpropagating laser pulse is reflected and upshifted in frequency with a multiplication factor of 37-66, corresponding to the extreme ultraviolet wavelength. The demonstrated flying mirror reflectivity (from 3 x 10(-6) to 2 x 10(-5), and from 1.3 x 10(-4) to 0.6 x 10(-3), for the photon number and pulse energy, respectively) is close to the theoretical estimate for the parameters of the experiment.