# 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 $x, y \in R$.

Then:

• $\norm {x + y} \lt \norm y \implies \norm x = \norm y$
• $\norm {x - y} \lt \norm y \implies \norm x = \norm y$
• $\norm {y - x} \lt \norm y \implies \norm x = \norm y$

## Proof

The contrapositive statements are proved.

Let $\norm x \ne \norm y$

By Corollary 2 then:

$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} \ge \norm y$

The result follows.

$\blacksquare$