Proof verification: Let $A, C \subseteq X$. Prove that $C - (C \cap A) = C \cap (X-A)$.

Question

Needing this statement for a topology problem - looking for proof verification here. I am always a bit uneasy when using the addition rule.

For the forward direction, suppose $x \in C - (C\cap A)$. Then $x \in C$ and either $x \notin C$ or $x \notin A$. Since $x \in C$ we conclude $x \notin A$. Since $A \subseteq X$ we conclude $x \in C \cap (X - A)$.

Conversely, let $x \in C \cap (X - A)$. Then $x \in C$ and $x \in X$ and $x \notin A$. Via addition, we may conclude $x \in C$ and, additionally, $x \notin C$ or $x \notin A$, which implies $x \in C - (C \cap A)$. QED

Answer 1

$$ \begin{aligned} C-(C\cap A)&=C\cap(X-(C\cap A))\\ &=C\cap((X-C)\cup (X-A))\\ &=(C\cap (X-C))\cup(C\cap (X-A))\\ &=C\cap (X-A) \end{aligned}$$