because His saying in this case includes the relatives of the Prophet with its generality, and therefore, the priority of the children of Fatimah over all Muslims and the rest of the relatives of the Prophet is proven in all matters in which priority has a role to play, such as the Caliphate, and therefore, any of them who is considered the elder of them in his time is more deserving of the Prophet and his works than others; such as Hasan after Ali ibn Abi Talib, Husain after Hasan, Ali ibn Husain (d. 94 AH) after Husain, Abu Ja‘far Muhammad ibn Ali known as al-Baqir (d. 114 AH) after Ali ibn Husain, and Ja‘far ibn Muhammad known as al-Sadiq (d. 148 AH) after Abu Ja‘far, each of whom was considered the elder of the Ahl al-Bayt in his time.

[Ali, Fatimah, Hasan, and Husain are the intended meaning of the Ahl al-Bayt of the Prophet]

Yes, what is meant by the Ahl al-Bayt of Prophet is originally Ali, Fatimah, Hasan, and Husain, and this is what is required by their closeness to the Prophet , their residing in his house, and the mutawatir reports narrated by many of his Sahabah, such as Abdullah ibn Abbas[1], Abu Sa‘id al-Khudri[2], Sa‘d ibn Abi Waqqas[3], Anas ibn Malik[4],