Action of Group on Coset Space is Group Action

Theorem
Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Then $G$ acts on the left coset space $G / H$ by the rule:
 * $\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$

Proof
demonstrating that $*$ satisfies.

Then:

demonstrating that $*$ satisfies.