Proof on set equality I was making a few exercises on set proofs but I met an exercise on which I don't know how to start:

If $A \cap C = B \cap C $ and $ A-C=B-C $ then $A = B$

Where should I start? Should I start from $ A \subseteq B $ or should I start from this $ ((A\cap C = B\cap C) \land  (A-C = B-C)) \Rightarrow (A = B)$ ? 

\begin{align}
A\cap C&=B\cap C\\
A-C&=B-C
\end{align}
Take the union of the left- and right-hand sides:
\begin{align}
(A\cap C)\cup(A-C)&=(B\cap C)\cup(B-C)\\
(A\cap C)\cup(A\cap\overline C)&=(B\cap C)\cup(B\cap\overline C)\\
A\cup(C\cap\overline C)&=B\cup(C\cap\overline C)\\
A&=B
\end{align}
