Equivalent definition of compact support modulo $H$

Question

My question is how to prove the equivalenc:

Given a function $f\in X$, the following definition for compact supported modilo H are equivalent:

1. The image of the support supp $f$ of $f$ in $H\backslash G$ is compact.

2. supp $f\subset HC$ for some compact set $C$ in $G$.

I found a similar question here that may help(link), but after reading that I still can't solve it myself.