Intersection of an approximate group power with a subgroup

If $A$ is a $K$-approximate group in $G$ and $H\le G$, then $A^m\cap H$ is a $K^{O(m)}$-approximate group. Indeed, the at most $K^{m-1}$ translates of $A$ covering $A^m$ induce at most that many translates of $A^2\cap H$ covering $A^m\cap H$, and the same argument covers its square inside $A^{2m}\cap H$.