Alternating Group is Simple except on 4 Letters/Lemma 2

Theorem
Let $n$ be an integer such that $n \ge 5$.

Let $A_n$ denote the alternating group on $n$ letters.

Let $\alpha \in A_n$ be the permutation on $\N_n$ in the form:
 * $\alpha = \tuple {1, 2} \tuple {3, 4}$

Let $\beta$ be the $3$-cycle $\tuple {3, 4, 5}$.

Then:
 * $\beta^{-1} \alpha^{-1} \beta \alpha = \beta$