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Proves that any (memory-less) multiple-access channel (MAC) comprises finite sub-MAC's in which the Kuhn-Tucker (KT) condition for each sub-MAC is necessary and sufficient. It means that the total capacity is evaluated for a T-user MAC with mathematical rigorousness. The KT-condition for the MAC is necessary, and is not in general sufficient for evaluating the total capacity. However, we could find out a set of sub-MAC's where the KT-condition is sufficient. To prove the sufficiency, use is made for the boundary equation regarding the capacity region of the sub-MAC.