Direct factor implies transitively normal

Statement
Suppose $$H$$ is a direct factor of a group $$G$$. Then, $$H$$ is a transitively normal subgroup of $$G$$. In other words, for any normal subgroup $$K$$ of $$H$$, $$K$$ is also normal in $$G$$.

Facts used

 * 1) uses::Direct factor implies central factor
 * 2) uses::Central factor implies transitively normal

Proof using given facts
The proof follows from facts (1) and (2).