Should the employee be bound by the contract concluded with the institution, which stipulates his work from seven in the morning until one in the afternoon, or by the word of the official who allows him to finish work at ten?

If the person in charge is authorized to grant permission for early departure, then there is no harm for your friend in doing so. However, if he is not authorized, then it is not permissible for your friend to leave even if the person in charge permits him, and he must adhere to the terms of the employment contract.