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:  The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace is a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors).