Product of Hausdorff Factor Spaces is Hausdorff/General Result

Theorem
Let $\mathbb S = \left\{{\left({S_\alpha, \tau_\alpha}\right)}\right\}$ be a set of topological spaces for $\alpha$ in some indexing set $I$.

Let $\displaystyle T = \left({S, \tau}\right) = \prod \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\mathbb S$.

Let each of $\left({S_\alpha, \tau_\alpha}\right)$ for $\alpha \in I$ be $T_2$ (Hausdorff) spaces.

Then $T$ is a $T_2$ (Hausdorff) space.