Factor Spaces of Hausdorff Product Space are Hausdorff

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 $T$ be a $T_2$ (Hausdorff) space.

Then:
 * for each $\alpha \in I$, $\struct{S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space.