Set Intersection Not Cancellable

Theorem
Let $$S$$ be a set and let $$\mathcal P \left({S}\right)$$ be the power set of $$S$$.

Let $$S_1, S_2, T \in \mathcal P \left({S}\right)$$.

Suppose that $$S_1 \cap T = S_2 \cap T$$.

Then it is not necessarily the case that $$S_1 = S_2$$.

Proof
Proof by counterexample:

Let $$S = \left\{{1, 2, 3}\right\}$$.

Let $$T = \left\{{3}\right\}$$.

Let $$S_1 = \left\{{1, 3}\right\}, S_2 = \left\{{2, 3}\right\}$$

Then $$S_1 \cap T = S_2 \cap T = \left\{{3}\right\}$$ but $$S_1 \ne S_2$$.