Odd Order Group Element is Square

Theorem
Let $$\left({G, \circ}\right)$$ be a group whose identity is $$e$$.

Let $$x \in G$$ such that $$\left|{x}\right|$$ is odd.

Then $$\exists y \in G: y^2 = x$$.

Proof
Let $$\left|{x}\right| = 2 n - 1$$. Then:

$$ $$ $$ $$

Hence the result.