# Combination Theorem for Continuous Mappings/Topological Division Ring/Translation 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:

$\forall x \in S: \map {\paren {\lambda + f} } x = \lambda + \map f x$

Then

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

## Proof

By definition of a topological division ring:

$\struct {R, +, *, \tau_{_R} }$ is a topological ring.
$\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.

$\blacksquare$