To implement Ginzburg's equality check procedure for regular expressions by using personal computers, we propose a new and more efficient axiom system consisting of an axiom and inference rules concerning a new relational symbol C in addition to a part of Salomaa's axiom system. The researches of axiom systems for the regular expressions started in the 60's. Redco [4] showed that it is impossible to make a complete axiom system with finite axioms and only one inference rule R1. He made a complete system with countably many axioms and an inference rule Rl. Salomaa [5] showed a complete and consistent axiom system by using two inference rules RI and R2. In his system every tautology has a constructive proof. But the construction of a proof is so complicated that it is not useful for practice. After a few years, Ginzburg [2] showed a simple mechanical procedure for checking equality of regular expressions by using derivatives. This method uses transition graphs so it is neither symbolic nor axiomatic. In the present paper, by using symbolic derivatives, we improve Ginzburg's procedure to a symbolic and axiomatic one. Moreover we examine which axiom in Salomaa's system is essen tial for showing the validity of the procedure. We choose some axioms which is useful for mechanical procedure, and we show that it is sufficient to show the termination of the checking procedure by using only those axioms. In section 2, we recall Salomaa's system Fi and introduce a new system F. In section 3, we show that in the system F*, the set of all symbolic derivatives of a regular expression is finite. This result guarantees the termination of improved Ginzburg's equality check procedure. In section 4, we introduce a new relational symbol C, and we improve the system F* to a more efficient system F+. Since we do not use an axiom of associative laws in the system F+, we can check equalities by a straightforward method. We implemented the efficient procedure on a personal computer. So we can easily check the equality of two regular expressions. Moreover we get the transition graphs which accept the expressions together with the answer of the equality. We show a brief example in the Appendix.