How do I prove that a Euclidean function induces a submultiplicative Euclidean function

Question

Let $R$ be a Euclidean Domain and $g:R\setminus\{0\}\rightarrow \mathbb{N}$ be a function such that for each $a,b\in R (b\neq 0)$ there exists $q,r\in R$ such that $a=bq+r$ and $g(r)<g(b)$ or $r=0$.

Now define $f(x)=\min_{x\in R\setminus\{0\}} g(xa)$ for all $x\in R\setminus \{0\}$.

Why is this $f$ a Euclidean function?

Answer 1

See Proposition 4 of the following article:

• Pierre Samuel, About Euclidean Rings, link