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

Theorem
Let $\struct{S, \tau_S}$ be a topological space.

Let $\struct{R, +, *, \tau_R}$ be a topological division ring.

Let $f,g : \struct{S, \tau_S} \to \struct{R, \tau_R}$ be continuous mappings.

Let $f * g : S \to R$ be the mapping defined by:
 * $\forall x \in S: \map {\paren{f * g}} x = \map f x * \map g x$

Then
 * $f * g : \struct{S, \tau} \to \struct{R, \tau_R}$ is continuous.

Proof
By definition of a topological division ring, $\struct{R, +, *, \tau_R}$ is a topological ring.

From Product Rule for Continuous Mappings to Topological Ring, $f * g : \struct{S, \tau_S} \to \struct{R, \tau_R}$ is a continuous mapping.