Proposition 1

Let $v$ be a valuation on a (not necessarily commutative) ring $C$. For any elements $x_i \in C$ ($1 \leq i \leq n$),

Moreover, if there exists a single index $k$ such that $v(x_k) = \inf_{1 \leq i \leq n} v(x_i)$, the two sides of (1) are equal. In particular, if $v(x) \neq v(y)$, then $v(x + y) = \inf(v(x), v(y))$.

Relation (1) follows from axiom (VL_{II}) by induction on $n$. If there exists a single index $k$ such that $v(x_k) = \inf_{1 \leq i \leq n} v(x_i)$, then, writing $y = \sum_{i \neq k} x_i$ and $z = \sum_{i=1}^n x_i$, $v(y) > v(x_k)$ and $v(z) \geq v(x_k)$ by (1); if $v(z) > v(x_k)$, the relation $x_k = z - y$ would give $v(x_k) \geq \inf(v(z), v(y)) > v(x_k)$, which is absurd; whence $v(z) = v(x_k)$, which proves the second assertion.