Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule

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

Let $\struct{G, \circ, \tau_G}$ be a topological semigroup.

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

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

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