In the definition of valuation I have, the axioms are:
$val: K \rightarrow \mathbb{R} \cup \infty$, $K$ is a field
$val(a) = \infty \Leftrightarrow a = 0$
$val(ab) = val(a) + val(b)$
$val(a+b) \geq min(val(a), val(b))$
In other texts, I have seen a strengthening of the second rule
$val(a) \neq val(b) \Rightarrow val(a+b) = min(val(a), val(b))$
I'm struggling to prove this from my axioms, any hints will be greatly appreciated