Thought experiments are widely used in informal explanations of Relativity Theories; however, they are not present explicitly in formalized versions of Relativity Theory. In this paper, we present an axiom system of Special Relativity which is able to express thought experiments formally and explicitly. Moreover, using these thought experiments, we can provide an explicit definition of relativistic mass based merely on kinematical concepts and thought experiments on collisions. Using this definition, we can geometrically prove the Mass Increase Formula (m_0 = m mathrel {cdot }sqrt{1-v^2}) in a natural way, without postulates of conservation of mass and momentum.