Intersection of DPICF and direct factor is central factor

Verbal statement
The intersection of a direct projection-invariant central factor with a direct factor is a central factor.

Hands-on proof
Suppose $$H$$ is a direct projection-invariant central factor of $$G$$, and $$K$$ is a direct factor of $$G$$. We need to show that $$H \cap K$$ is a central factor of $$G$$.

Since the property of being a central factor is transitive, and $$K$$ is already a central factor, it sufices to show that $$H \cap K$$ is a central factor of $$K$$.

In order to do this, consider a direct projection $$\pi:G \to K$$ corresponding to the direct factor $$K$$. Since $$H$$ is direct projection-invariant, it must map to inside $$H \cap K$$. Further, the elements of $$H \cap K$$ are $$\pi$$-invariant, and hence the image of $$H$$ under $$\pi$$ is in fact $$H \cap K$$.

Now, since $$H$$ is a central factor of $$G$$, $$\pi(H)$$ is a central factor of $$\pi(G)$$, telling us that $$H \cap K$$ is a central factor of $$K$$ and we are done.