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A method called spectrum analysis for a software requirements specification enables us to identify the completeness and the correctness of the specification. The method can be systematically applied when the specification is written in a semi-formal notation based on the semantic structure of each sentence in the specification. By using the rules based on the semi-formal notation, necessary quality requirements descriptions for each sentence can be recommended. However, the case studies for evaluating the rules have not been fully achieved. In addition, how to concretely update the requirements specification according to the recommendation rules is not defined. In this paper, we report the case study to explore how to improve a concrete specification based on the results of spectrum analysis. Such an exploration can be a basis for improving the way of defining the rules. As a result, the way of defining the rules seems to be reasonable, but the organization among the elements of the rules such as the concepts and types should be introduced to facilitate the systematic application of the rules. In addition, the current semi-formal notation should be extended so as to explicitly represent the adverbs, which are useful for specifying quality requirements.