Normal implies modular

Statement
Suppose $$A$$ is a normal subgroup of $$G$$. Then, for any subgroup $$B$$ of $$G$$ and any subgroup $$C$$ of $$G$$ such that $$A \le C$$, we have:

$$A(B \cap C) = AB \cap C$$

Facts used

 * 1) uses::Normal implies permutable
 * 2) uses::Permutable implies modular (which in turn uses uses::modular property of groups)

Proof
The proof follows directly by combining facts (1) and (2).