Three Points in Ultrametric Space have Two Equal Distances/Corollary 3

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,

Let $x, y \in R$ and $\norm x \lt \norm y$.

Then:
 * $\norm {x + y} = \norm y$

Proof
By Corollary 2 then:
 * $\norm{x + y} = \max \set {\norm x, \norm y} = \norm y$