Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule

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

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

Let $\lambda \in G$.

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

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

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

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