Corollary 1

For a mapping $f$ from $K$ to $\mathbf{R}_+$ to be an absolute value, it is necessary and sufficient that it satisfy conditions $(\mathbf{VA}_I)$, $(\mathbf{VA}_{II})$ and $(\mathbf{U}_2)$.

It is necessary, for $(\mathbf{VA}_{II})$ implies

Conversely, suppose $f$ satisfies $(\mathbf{VA}_I)$, $(\mathbf{VA}_s)$ and $(\mathbf{U}_2)$; for every integer $n > 0$, let $r$ be the least integer such that $2^r \geq n$; if in (1) $A$ is replaced by 2, the $x_i$ of index $i \leq n$ by 1 and the $x_i$ of index $i > n$ by 0, we obtain

then Proposition 2 may be applied with $C = 2$ and hence $f$ is an absolute value.