Product of Row Sum Unity Matrices

Theorem
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix.

Let $\mathbf B = \sqbrk b_{n p}$ 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:


 * $\ds \sum_{i \mathop = 1}^n a_{i j} = \sum_{i \mathop = 1}^n b_{i j} = 1$

Then: