Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule

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 element 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.