I derive an equation and get the term $\sum\limits_k \frac{W_{ik} H_{kj}}{\sum\limits_k W_{ik} H_{kj}}$. I think this term is equal to 1 because
Proof:
$\sum\limits_k \frac{W_{ik} H_{kj}}{(\sum\limits_k W_{ik} H_{kj})_{ij}} = \frac{1}{(\sum\limits_k W_{ik} H_{kj})_{ij}} \times \sum\limits_k W_{ik} H_{kj} = 1$
where $W$ and $H$ are matrices with dimension $i \times k$ and $k \times j$, respectively
I just wonder is this proof correct?