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

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## 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

- $\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
- $f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ 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.

$\blacksquare$