Proportion of Power

Theorem
Let $x$ and $y$ be proportional.

Let $n \in \Z$.

Then $x^n \propto y^n$.

Proof
Let $x \propto y$.

Then $\exists k \neq 0: x = k \times y$ by the definition of proportion.

Raising both sides of this equation to the $n$th power:

so $k^n$ is the desired constant of proportion.

The result follows from the definition of proportion.