Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff

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

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

Then $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.