Abstract. Let w(x) be a one-variable equation in a free group F of finite rank. Lyndon has proved that it is possible to associate effectively to w(x) the set of its solutions, whereas Appel and Lorenc have provided a simpler representation of the set inferred. In this paper, we invert the problem and demonstrate that if the elements of any set S c F are solutions of an equation w(x), then w(x) belongs to the normal closure of finitely many short equations associated to S. A few consequences are given.
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