Symmetric Difference is Associative/Proof 1

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 RHS:

Expanding the LHS:

From Union is Commutative it is seen that the LHS and RHS are the same, and the result is proved.