Union of Power Sets not always Equal to Powerset of Union

Theorem
The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union.

Proof
Proof by Counterexample:

Let $S = \set {1, 2, 3}, T = \set {2, 3, 4}, X = \set {1, 2, 3, 4}$.

But note that $X \nsubseteq S \land X \nsubseteq T$.

Thus:

So:
 * $\powerset {S \cup T} \ne \powerset S \cup \powerset T$