With Another Authority

I don't understand the usage of the preposition 'with' on some website:

http://www.hacsc.org/p_programs.php
"The Section 8 Housing Choice Voucher Program provides rental subsidies for very low-income households who reside in privately owned rental units and pay 30% of their income towards rent. The balance of the rent is paid by the Housing Authority directly to the owner. The subsidy is tied to the family so if the family moves to another residence, the subsidy moves with them. Participants can also choose to exercise a portability feature that allows them to move to another county or state and continue to use their subsidy with another housing authority. "

I checked this dictionary:

http://www.learnersdictionary.com/search/with

but couldn't find a good definition. Would this be better?

"use their subsidy under the jurisdiction of another housing authority"