We presents the results of study of focusing and imaging properties of double-lens system for hard x-ray radiation consisting of two Fresnel zone plates (ZP) made from silicon. We demonstrate for the first time the phenomenon of focusing by two crystal ZP located at significant distance from each other. We investigate by both theoretically and experimentally the peculiarities of intensity distribution at the focal plane during a scan by second ZP normally to the optical axis of the system. We investigate as well the intensity distribution along the optical axis for our double-lens system from crystal ZP. We realize experimentally a registration of the focused image of the object by means of double-lens system based on ZP. Measurements are performed on the beam line BM-5 of the European Synchrotron Radiation Facility (ESRF) at the x-ray energy 9.4 keV. We elaborate a computer program for theoretical simulation of the optical properties of x-ray double-lens system based on ZPs. A calculation is made by convolution of transmission function and Kirchhoff propagator in paraxial approximation by means of Fast Fourier Transformation.
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