whose column vectors are the two positive P$_{j}$ α . For each I$_{i}$ , we define which is a ∏ $^{j}$∈I$_{i}$ k$_{j}$ by 2 n$_{i}$ matrix. n$_{i}$ is the number of items in I$_{i}$ . Each P$_{j}$ is a full column rank matrix of rank 2. Thus, P i is rank of 2 n$_{i}$ matrix and is a full column rank matrix. Each column vector of T$_{I}$$_{i}$ is a column vector of P $_{i}$. We can show that for two classes α 1 = α $_{2}$, α $_{1}$-th and α $_{2}$-th columns of T$_{I}$$_{i}$ are not identical. We prove this by contradiction. Suppose that they are the same. It is easy to see that the α $_{l}$-th column in T$_{I}$$_{i}$ has the form l = 1 , 2. So P i = ⊗ $^{j}$∈I$_{i}$ P$_{j}$ ⊗ $^{j}$∈I$_{i}$     p 1 j $^{α}$l p 2 j $^{α}$l . p k$_{j}$ j $^{α}$l    $^{,}$ ⊗ $^{j}$∈I$_{i}$     p 1 j $^{α}$1 p 2 j $^{α}$1 . p k$_{j}$ j $^{α}$1     = ⊗ $^{j}$∈I$_{i}$     p 1 j $^{α}$2 p 2 j $^{α}$2 . p k$_{j}$ j $^{α}$2    $^{.}$ (11) However, we can find item j ∗ ∈ I$_{i}$ such that for all k = 1 , ..., k$_{j}$ ∗ − 1 which means p 1 j $^{∗$_{α}$}$1 + ... + p k j $^{∗$_{α}$}$1 = p 1 j $^{α}$2 + ... + p k j $^{∗$_{α}$}$2     p 1 j $^{∗$_{α}$}$1 p 2 j $^{∗$_{α}$}$1 . p k$_{j}$ ∗ j $^{∗$_{α}$}$1     =     p 1 j $^{∗$_{α}$}$2 p 2 j $^{∗$_{α}$}$2 . p k$_{j}$ ∗ j $^{∗$_{α}$}$2    $^{.}$ 37