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

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

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

Let $\lambda \in R$.

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

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

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

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

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

From Multiple Rule for Continuous Mappings to Topological Ring, $\lambda * f, f * \lambda : \struct{S, \tau_S} \to \struct{R, \tau_R}$ are continuous mappings.