Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $U \subseteq S$ be an open set in $T$.

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

Let $f: U \to R$ and $g: U \to R$ be continuous mappings.

Then:
 * $f g : U \to R$ is continuous.

where $f g : U \to R$ is the mapping defined by:
 * $\forall x \in U: \map {\paren{f g}} x = \map f x \circ \map g x$