Order of Conjugate Element equals Order of Element/Corollary

Corollary to Order of Conjugate Element
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Then:
 * $\forall a, x \in \left({G, \circ}\right): \left|{x \circ a}\right| = \left|{a \circ x}\right|$

where $\left|{a}\right|$ is the order of $a$ in $G$.

Proof
From Order of Conjugate Element, putting $a \circ x$ for $a$:
 * $\left|{x \circ \left({a \circ x}\right) \circ x^{-1}}\right| = \left|{a \circ x}\right|$

from which the result follows by $x \circ x^{-1} = e$.