# Combination Theorem for Continuous Mappings/Topological Division Ring/Product 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 Product Rule for Continuous Mappings to Topological Ring, $f * g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.

$\blacksquare$