Product of Row Sum Unity Matrices

Theorem
Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix.

Let $\mathbf B = \left[{b}\right]_{n p}$ be an $n \times p$ matrix.

Let the row sum of $\mathbf A$ and $\mathbf B$ be equal to $1$.

Then the row sum of their (conventional) product is also $1$.

Proof
We have that:


 * $\displaystyle \sum_{i \mathop = 1}^n a_{ij} = \sum_{i \mathop = 1}^n b_{ij} = 1$

Then: