# Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule

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## Theorem

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

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

Let $\lambda, \mu \in R$ be arbitrary elements in $R$.

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

Let $\lambda * f + \mu * g: S \to R$ be the mapping defined by:

- $\forall x \in S: \map {\paren {\lambda * f + \mu * g}} x = \lambda * \map f x + \mu * \map g x$

Let $f * \lambda + g * \mu : S \to R$ be the mapping defined by:

- $\forall x \in S: \map {\paren {f *\lambda + g * \mu}} x = \map f x * \lambda + \map g x * \mu$

Then:

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

## Proof

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