Product of Metacompact Spaces is not always Metacompact

Theorem
Let $I$ be an indexing set.

Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

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

Let each of $\struct {S_\alpha, \tau_\alpha}$ be a metacompact space.

Then it is not necessarily the case that $\struct {S, \tau}$ is also metacompact space.

Proof
Let $T$ be the Sorgenfrey line.

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

From Sorgenfrey Line is Metacompact Space, $T$ is a metacompact space.

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

Hence the result.