Three Points in Ultrametric Space have Two Equal Distances/Corollary 4

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

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

Then:
 * $\norm x = \norm y$

Proof
By Corollary 3 then:
 * $\norm x = \norm {\paren {x - y} + y} = \norm y$