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Testing is one important means to improve reliability and quality of software systems. The testing for object-oriented software systems is more difficult than that of procedure-oriented programming. Based on the concept of equivalent fundamental pairs, we have proposed a new methodology for testing object-oriented software at the class level. The main results of the methodology are published in ACM Transactions on Software Engineering and Methodology (see vol.7, no.3, p.250-95, 1998). The paper refines some algorithm in the methodology. Some problem in the methodology seems undecidable at the first glance. However, the refinement in this paper shows that it is decidable.