  By (a), this set is pairwise orthogonal. Now let \(\alpha\) be the state on \(\mathcal{A}_5\) taking the value 1/2 on outcomes \(a, b, c, d\) and \(e\), and the value 0 on \(x, y, z, w\) and \(v\). By condition (b), there exists state \(\omega\) on \(\mathcal{B}\) such that 