Before revision:   The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace can be written as a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors).
After revision:  However, only a decomposition without cancelations is biochemically meaningful, since a reversible reaction cannot have different directions in the contributing elementary modes. This essential requirement has been largely overlooked by the metabolic pathway community. Indeed, every flux mode can be decomposed into elementary modes without cancelations. The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace can be written as a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors).