\textit{A. Gibbard} [Econometrica 41, 587-601 (1973; Zbl 0325.90081)] and \textit{M. A. Satterthwaite} [J. Econ. Theory 10, 187-217 (1975; Zbl 0315.90088)] have independently proved that if a social choice function or voting scheme f is strategy-proof (or non-manipulable) and the range of f contains at least three alternatives, then it is dictatorial. In the present paper a new proof of this theorem is presented. The author's approach focuses on the local distribution of power among single individuals, i.e., on the ability of each single voter to determine the social outcome, given the preferences of all other individuals in society. An individual is called a ''pivotal voter'' at a certain profile if he has at least one way of changing the social outcome by alternating his vote. It is shown in the proof that there must be at least one preference profile where some individual is pivotal for more than two alternatives. Then, however, this individual turns out to be a dictator. The author has used his concept of pivotal voters also to present a new proof of Arrow's Impossibility Theorem [Economic Letters 6 (1980)].