Corollary 2

For a mapping $f$ of $K$ to $\mathbf{R}_+$ to belong to $\mathscr{V}(K)$, it is necessary and sufficient that it be of the form $g^t$, where $t > 0$ and $g$ is an absolute value on $K$.

To say that $f$ satisfies $(\mathbf{U}_A)$ is equivalent to saying that $f^s$ satisfies $(\mathbf{U}_{A^s})$; as there exists $s > 0$ such that $A^s \leq 2$, Corollary 1 shows that for such a value of $s$, $f^s$ is an absolute value.