We want to show that :

Part I: TPT:

Proof of Part I: let

Then, . By definition of symmetric difference,

Hence, , , OR ,

That is, .

Hence, , and

Hence, and .

Hence, .

Hence, , , but .

Therefore, , but . —- Call this I.

Consider .

Therefore, .

Therefore, , and

Therefore, , and .

Therefore, .

Hence, , , but .

That is, , . Call this II.

From I and II, .

Part II: TPT: .

Quite simply, reversing the above steps we prove part II.

QED.

Cheers,

Nalin Pithwa