This is what I have so far:
First, we will attempt to show $A-(B-C) \subseteq (A-C) - C.$ Let $x \in A -(B-C)$. Then $x\in A$ and $x \notin (B-C)$. By DeMorgan's we have that $$x \notin (B-C) = x \notin B \vee x \in C.$$ We have that $x \in A \wedge (x \notin B \vee x \in C)$. Then we have that $(x \in A \wedge x \notin B) \vee (x \in A \wedge x \in C)$.
So by definition of $\cup$ and $\cap$, we have $x \in (A-B) \cup (x\in (A \cap C))$. I'm kind of stuck here.
I can see a contradiction here in that LHS says $x \in C$ but the RHS says $x \notin C$.
Can anyone tell me what to do from here? I'm not quite sure how to formally state that the statement is false. I feel like I'm not being complete here.