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The matter of Specific Relativity axiomatization has long been the attention of physicists and logicians. But owing to the problem of cross disciplinarity, the issue has never found a proper solution. The Hungary scholars Hajnal Andréka et al put forward a first-order logic system termed Specrel about the Specific Relativity in 2007. But the system is insufficient in depicting the Specific Relativity either in logic or physics. Against that background, the present paper proposes another logic system, Specrel', and by that system tries to prove the theory of Lorentz Transformation, the fundamental formula in Specific Relativity, which can hence obtain such exemplar effects as time dilatation and length contraction. [ABSTRACT FROM AUTHOR]