Product of Hausdorff Factor Spaces is Hausdorff/General Result

Theorem
Let $\SS = \family{\struct{S_\alpha, \tau_\alpha}}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$.

Let $\displaystyle T = \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\SS$.

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

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