# Combination Theorem for Continuous Mappings/Topological Division Ring/Sum 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 $f, g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ 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

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

## Proof

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

From Sum Rule for Continuous Mappings into Topological Ring:

- $f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.

$\blacksquare$