This paper is concerned with undirected graphs which may be infinite and may contain multiple edges. The Axiom of Choice is assumed. The terms path, infinite path and circuit are used in the same sense as Weg, unendlicher Weg and Kreis, respectively, are used in D. Konig's book [1]. The valency of a vertex is the number of edges incident with it.The length of a path is the number of edges in it. The following theorem is a generalization of the well known fact that if a vertex of a graph is not a cut-vertex (Artikulation [2]) and has valency ≧2, then the graph contains at least one circuit to which the vertex belongs.