Let $R$ be a UFD with a "normalization" function $N: R \to R$ so that
$$(a \sim b \implies N(a)=N(b)), N(a) \sim a$$
Then define the leading unit $lu(a)$ as the unique $u$ so that $a=uN(a)$ if $a$ is non-zero and $1$ if $a$ is $0$.
Prove that $pp(a)$ is "normalized" which means $lu(pp(a))=1$. I only know that $pp(a)=\frac{a}{cont(a)}$ where $cont(a)$ is the content of $a$ but I don't know how to proceed further. Essentially I need to prove $pp(a)=N(pp(a))$ because then the leading unit is $1$? If it helps I have already proved that $lu(ab)=lu(a)lu(b)$ for non-zero $a,b$.