Symmetric Difference is Associative

Theorem
Symmetric difference is associative:

$$R * \left({S * T}\right) = \left({R * S}\right) * T$$

Proof
We can directly expand the expressions for $$R * \left({S * T}\right)$$ and $$\left({R * S}\right) * T$$, and see that they come to the same thing.

Expanding the LHS:

Expanding the RHS:

Thus we see that (by applying the fact that Union is Commutative) the LHS and RHS are the same, and the result is proved.