Chapter 2.2, Problem 22E

Let $x$ and $y$ be integers, and let $m$ and $n$ be positive integers. Use mathematical induction to prove the statements in Exercises $18-23$. $($ The definitions of $x^n$ and $nx$ are given before Theorem $2.5$ in Section $2.1$ $)$

$\mathrm{}\left(m+n\right)x=mx+nx$