Euler Phi Function of Non-Square Semiprime/Proof 2

Proof
A semiprime with distinct prime factors is a square-free integer.


 * $\displaystyle \phi \left({n}\right) = \prod_{\substack {p \mathop \backslash n \\ p \mathop > 2} } \left({p - 1}\right)$

where $p \mathrel \backslash n$ denotes the primes which divide $n$.

As there are $2$ prime factors: $p$ and $q$, this devolves to:
 * $\displaystyle \phi \left({n}\right) = \left({p - 1}\right) \left({q - 1}\right)$

except when $p = 2$, in which case:
 * $\displaystyle \phi \left({n}\right) = q - 1$

But when $p = 2$, we have that $p - 1 = 1$ and so:
 * $\left({p - 1}\right) \left({q - 1}\right) = q - 1$

Hence the result.