I've just started learning information theory and have stumbled upon a roadblock due to my apparently not understanding how summation works. I do not understand why this is true:
$- \sum\limits_{i,j}^{} p(x_i, y_j) \log p(y_j | x_i) - \sum\limits_{i,k}^{} p(x_i, z_k) \log p(z_k | x_i) = - \sum\limits_{i,j,k}^{} p(x_i, y_j, z_k) [ \log p(y_j | x_i) + \log p(z_k | x_i) ]$
Can someone explain the justification for this equality? My book is less than forthcoming and I'm not even sure what to search for.