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For the identity, we will show $A\cap (B - C) \subseteq (A\cap B) - (A\cap C)$ and $A\cap (B - C) \supseteq (A\cap B) - (A\cap C)$.

Let $x\in A\cap (B - C)$.

Then $x\in A$ and $x\in B - C$.

So $x\in A$ and $x\in B$ and $x\notin B\cap C$.

Is this the right approach? I know $B-C = $ some other expression but I can't remember it.