Clarification on Metric Induced by Norm is Symmetric

Question

On the ProofWiki page for showing a Metric Induced by Norm is Metric under the "Proof of M3", is the claim that $ |-1| \times ||y - x|| = ||y - x|| $. I do not understand how this step follows, in particular, how $ |-1| = 1 $. This step is actually clear in the usual $ \mathbb{R} $ or $ \mathbb{C} $, but how does this follow from an arbitrary normed division ring which ProofWiki uses in its definition for a normed vector space? I could only manage to prove $ |-1| $ must equal $1$ or $-1$ and the related fact that $|1| = 1$, but not that $|-1| = 1$ which is a key step in the proof.

Answer 1

If you just look at the ProofWiki article on normed division rings, the answer is right there:

A norm on [a division ring] $R$ is a map from $R$ to the non-negative reals [...]