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# Integer multiplication is strictly increasing

We must prove that, for three integers $n,m, p \in \mathbb{N}$, $n < m$, we have

$$n \times p < m \times p$$

## By Peano's axioms

The ordering in Peano's axioms is defined by, for $n,m \in \mathbb{N}$, we have $n \leq m$ if and only if there exists some integer $a \in \mathbb{N}$ such that

$$n + a = m$$

Therefore, we also get

\begin{eqnarray} m \times p &=& (n + a) \times p\\ &=& n \times p + a \times p \end{eqnarray}

Therefore, there exists an integer, $a \times p$, such that $m \times p = n \times p + a \times p$, so that

$$n \times p < m \times p$$