How does the situation of joint ownership in manumission differ from a situation where a slave owes a debt to two creditors, and one is paid?

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If the debt is established through a single cause, when one creditor is paid, the other creditor claims reimbursement from the paying creditor, similar to the manumission case. This differs from a general debt because the master's right is tied to the assets held by the partially freed slave, whereas a debt relates only to the debtor's personal liability (dhimmah) and not directly to external property held by a co-creditor.