Product of Paracompact Spaces is not always Paracompact

Theorem
Let $I$ be an indexing set.

Let $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \left({S, \tau}\right) = \prod_{\alpha \mathop \in I} \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$.

Let each of $\left({S_\alpha, \tau_\alpha}\right)$ be a paracompact space.

Then it is not necessarily the case that $\left({S, \tau}\right)$ is also paracompact space.

Proof
Let $T$ be the real number line with the right half-open interval topology.

Let $T' = T \times T$ be Sorgenfrey's half-open square topology.

From Right Half-Open Interval Topology is Paracompact Space, $T$ is a paracompact space.

From Sorgenfrey's Half-Open Square Topology is Not Paracompact Space, $T'$ is not a paracompact space.

Hence the result.