We want to add repeatedly.
\[ p_m(0) = 0 \quad \text{ and } \quad p_m(\ssuc{n}) = \ssuc{(p_m(n))} + m \]
for every natural number $n$.Let $m$ and $n$ be natural numbers. The value $p_m(n)$ is the product of $m$ with $n$.
We denote the product $p_m(n)$ by $m \cdot n$. We often drop the $\cdot $ and write $m \cdot n$ as $mn$.
The properties of products are direct applications of the principle of mathematical induction (see Natural Induction).2
\[ (k \cdot m) \cdot n = k \cdot (m \cdot n). \]
\[ m \cdot n = n \cdot m. \]