The present paper studies the problem of finding a two-layered perceptron that exactly classifies a given subset. Such a two-layered perceptron is called exact with respect to the given subset. We derive both a necessary and a sufficient condition for a given subset to be classifiable by an exact two-layered perceptron. The necessary condition can be viewed as a generalization of the linear-seperability condition of the one-layered perceptron and confirms the conjecture that the capabilities of exact two-layered perceptrons are more limited than those of exact three-layered perceptrons. The sufficient condition shows that the capabilities of exact two-layered perceptrons extend beyond the exact classification of convex subsets. Furthermore, we present a systematic verification method for the given sufficient condition. Keywords: Classification, Multi-Layered Perceptrons,Size Hidden Layer.