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The multiterminal hypothesis testing against is considered where and are separately encoded at rates and , respectively. The problem is to determine the minimum of the second kind of error probability, under the condition that the first kind of error probability for a prescribed . A good lower bound on the power exponent is given and several interesting properties are revealed. The lower bound is tighter than that of Ahlswede and Csiszár. Furthermore, in the special case of testing against independence, this bound turns out to coincide with that given by them. The main arguments are devoted to the special case with corresponding to full side information for . In particular, the compact solution is established to the complete data compression cases, which are useful in statistics from the practical point of view.