What is the ruling on the validity of a *Kitabah* contract where the *Mukatab* is required to pay a specified amount monthly, with immediate manumission upon paying the first installment?

Primary text

If a *Mukatab* contracts for a total sum, paying a fixed amount monthly, and it is stipulated that he is manumitted upon paying the first installment, the contract is valid according to the principal ruling of the madhhab, and manumission occurs upon that first payment. This is analogous to a situation where the master could have freed him without payment; thus, freeing him upon partial payment is permissible, leaving the remainder as a debt upon the now-freed man, similar to if he had sold himself to his master.