Normed Division Ring Operations are Continuous/Corollary

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

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$. Let $\tau$ be the topology induced by the metric $d$.

Then:
 * $\struct {R, \tau}$ is a topological division ring.

Proof
Since Addition is Continuous and Negation is Continuous, by the definition of a topological group then $\struct {R, +, \tau}$ is a topological group.

Since $\struct {R, +, \tau}$ is a topological group and Multiplication is Continuous, by the definition of a topological ring then $\struct {R, +, \times, \tau}$ is a topological ring.

Since $\struct {R, +, \times, \tau}$ is a topological ring and Inversion is Continuous, by the definition of a topological division ring then $\struct {R, +, \times, \tau}$ is a topological division ring.