Combination Theorem for Continuous Mappings/Topological Group/Product Rule

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

Let $\struct{G, *, \tau_G}$ be a topological group.

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

Let $f * g : S \to G$ be the mapping defined by:
 * $\forall x \in S: \map {\paren{f * g}} x = \map f x * \map g x$

Then:
 * $f * g : \struct{S, \tau_S} \to \struct{G, \tau_G}$ is a continuous mapping.

Proof
By definition, a topological group is a topological semigroup.

Hence $\struct{G, *, \tau_G}$ is a topological semigroup.

From Product Rule for Continuous Mappings to Topological Semigroup, $f * g : \struct{S, \tau_S} \to \struct{G, \tau_G}$ is a continuous mapping.