Stabilizer of Cartesian Product of Group Actions

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $S$ and $T$ be sets.

Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions.

Let the group action $*: G \times \left({S \times T}\right) \to S \times T$ be defined as:
 * $\forall \left({g, \left({s, t}\right)}\right) \in G \times \left({S \times T}\right): g * \left({s, t}\right) = \left({g *_S s, g *_T t}\right)$

Then the stabilizer of $\left({s, t}\right) \in S \times T$ is given by:
 * $\operatorname{Stab} \left({s, t}\right) = \operatorname{Stab} \left({s}\right) \cap \operatorname{Stab} \left({t}\right)$

where $\operatorname{Stab} \left({s}\right)$ and $\operatorname{Stab} \left({t}\right)$ are the stabilizers of $s$ and $t$ under $*_S$ and $*_T$ respectively.

Proof
By definition, the stabilizer of an element is defined as:
 * $\operatorname{Stab} \left({x}\right) := \left\{{g \in G: g * x = x}\right\}$

where $*$ denotes the group action.

So: