Suppose that $G$ is a locally comact group, let $H$ be a open subgroup of $G$ and let $\phi:G\to\Bbb C$ be a continuous function such that $Supp\phi=:\overline{\{x\in G: \phi(x)\neq 0\}}$ is compact. Prove that there exist a subset $\{x_1,x_2,\cdots, x_n\}\subset G$ such that $supp\phi\subset\cup_{i=1}^nHx_i$ and for any $i\neq j$ we have $Hx_i\cap Hx_j=\emptyset$.
Of course the first part is obvious because $G=\cup_{x\in G}Hx$ and because $Supp\phi\subset G$ is compact so there are $y_1,y_2,\cdots, y_m\in G$ such that $Supp\phi\subset\cup_{i=1}^mHy_i$. But How can I make them disjoint?
Hint: Recall that $G$ is partitioned into a disjoint union of cosets of $H$.