Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring

Theorem
Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.

Let $a \in R$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-closed ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$.

Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-open ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$.

Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-sphere of $a$ in $\struct {R, \norm {\,\cdot\,} }$.

Then:
 * $\map {S_\epsilon} {a; \norm {\,\cdot\,} } = \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} } \setminus \map {B_\epsilon} {a; \norm {\,\cdot\,} }$

Proof
The result follows directly from:
 * Closed Ball in Normed Division Ring is Closed Ball in Induced Metric
 * Open Ball in Normed Division Ring is Open Ball in Induced Metric
 * Sphere in Normed Division Ring is Sphere in Induced Metric
 * Sphere is Set Difference of Closed Ball with Open Ball