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This paper presents a new result generalizing Richards' theorem. It is shown that this result leads to a complete, simple and unified theory of cascade synthesis which yields the types A, B, Brune, C and D sections in a direct and natural manner. The element values of the various sections are obtained in closed form in terms of three or six indexes. Thus the extraction cycle is performed once and for all for the whole class of positive-real functions. Several problems are worked out in detail and a chart is constructed to facilitate the computations. The formulas are easily programmed on a digital computer.