A four-aspherical mirror system with unit magnification is investigated for use in deep ultraviolet(DUV) optical lithography. It is derived from the solution of all zero third order aberrations for the four-spherical mirror system with unit magnification. We have first examined the holosymmetric four-spherical mirror system in which all orders of coma and distortion are zero, and obtained a uniquely exact solution of all zero third order aberrations for the system. This system has no free design parameter left and has too rigid symmetry to correct the residual spherical aberration. To avoid lacks of optical design freedom of the holosymmetric system, new numerical solutions of non-holosymmetric system with all zero third order aberrations are derived in 1-dimensional design parameter domain. For the numerical solutions, several system characteristics such as possibility of fabrication, system size, vignetting, residual higher order aberrations are investigated and a system with the best characteristics is selected. In this system telecentricity in image space is obtained by stop shift. To reduce the residual higher aberrations of the selected system, aspherizations on the spherical surfaces are carried out. The computer-aided optimization technique gives a aspherical system with diffraction limited performance for DUV wavelength of $0.248 \mu m$(KrF excimer laser line). This system has N.A. of 0.4, image field diameter of 5.2mm, and relative radius of central obscuration of 0.60. Spot diagrams show Rayleigh resolution of $0.38 \mu m$ and depth of focus(DOF) of $1.0 \mu m$. MTF analysis with the central obscuration gives spatial frequency of 460cycles/mm at which MTF values for all fields are over than 0.45 within DOF of $1.0 \mu m$. In order to improve resolution keeping an appropriate depth of focus, the shorter wavelength is advantageous. So the new aspherization is optimized to give diffraction limited performance for the shorter DUV wavelength of $0.193 \mu m$(ArF excimer laser line). The final system we have obtained consists of all aspherized mirrors and has N.A. of 0.35, image field diameter of 5.2mm and relative radius of central obscuration of 0.58. Spot diagrams show Rayleigh resolution of $0.34 \mu m$ and DOF of $1.2 \mu m$. MTF analysis with the central obscuration gives spatial frequency of 540cycles/mm at which MTF values for all fields are over than 0.45 within DOF of $1.2 \mu m$. This reflective system is compact in size and expected to be useful in optical lithographic application.