Does Ahlul Bayt include the Nine descendants of Imam Husain(A.S.) ?

Another contributor commented that if based on Sunni references we quoted, Ahlul-Bayt are the Prophet, Fatimah, Ali, al-Hasan and al-Husain, then how can you include the nine descendants of alHusain(as)?
As we have already pointed out, only five members of Ahlul-Bayt were alive at the time of the Prophet(pbuh&hf). They were Prophet Muhammad, Lady Fatimah, Imam Ali, Imam al-Hasan, and Imam al-Husain. They were mentioned as Ahlul-Bayt by the Prophet(pbuh&hf) and the most important Sunni collections testify to this fact.
However the nine descendants of Imam al-Husain were not alive at that time so that the Prophet could not cover them by his mantle! But the Prophet did, in fact, mention their names and their numbers. Let us review some traditions from Sihah Sittah:
The Prophet(pbuh&hf) said: "al-Mahdi is one of us Ahlul-Bayt."
Sunni reference: Sunan Ibn Majah, v2, Tradition #4085
also:
The Prophet([buh&hf) said: "The Mahdi will be of my family, of the descendants of Fatimah (the Prophet's daughter).
Sunni references:
1. Sunan Abi Dawud, English version, Ch. 36, Tradition #4271 (narrated by Umm Salama, the wife of the Prophet)
2. Sunan Ibn Majah, v2, Tradition #4086
3. al-Nisa'i and al-Bayhaqi, and others as quoted in:
4. al-Sawa'iq al-Muhriqah, by Ibn Hajar al-Haythami, Ch. 11, section 1, p249
By the above traditions, the Prophet extended Ahlul-Bayt up to Imam al-Mahdi(as). So Ahlul-Bayt are not just those five, and Imam al-Mahdi is the last member of Ahlul-Bayt, but he was not born at the time of the Prophet so that he could take him into the Cloak as well! Also the messenger of Allah said:
"There shall be twelve Imams/Caliphs/Amirs for my nation"
Sunni references:
1. Sahih al-Bukhari, Arabic-English, v9, Tradition #329;
2. Sahih Muslim, English version, Chapter DCCLIV, v3, pp 1009-1010, Traditions #4476 --> #4483;
3. Sunan Abi Dawud, v2, p421 (three traditions);
4. Sahih al-Tirmidhi, v4, p501;
5. Musnad Ahmad Ibn Hanbal, v5, p106;
6. Others such as al-Tiyalasi, Ibn al-Athir, etc.
These 12 Imams will cover till the day of resurrection as Sahih Muslim testifies. The last of them is surely Imam al-Mahdi(as) who will appear in the last days and who is also from Ahlul-Bayt as the above tradition specified. There are other traditions in the Sunni collections in which the Prophet(pbuh&hf) has even mentioned the name of all these twelve individuals one by one.
"A Jew named al-A`tal came to the Prophet and said, "Muhammad! I wish to ask you about certain things which I have been keeping to myself; so, if you answer them, I shall declare my acceptance of Islam before you." The Prophet said, "Ask me, O father of Imarah!" So he asked him about many things till he was satisfied and admitted that the Prophet was right. Then he said, "Tell me about your wasi (successor): who is he? No prophet can ever be without a wasi; our prophet Moses had appointed Yusha` [Joshua] son of Noon as his successor." He said, "My wasi is Ali ibn Abu Talib followed by my grandsons al-Hasan and al-Husayn followed by nine men from the loins of al-Husayn." He said, "Then name them for me, O Muhammad!" The Prophet said, "Once al-Husayn departs, he will be succeeded by his son Ali; when Ali departs, his son Muhammad will succeed him. When Muhammad departs, his son Ja`far will succeed him. When Ja`far departs, he will be succeeded by his son Musa. When Musa departs, his son Ali will succeed him. When Ali departs, his son Muhammad will succeed him. When Muhammad departs, his son Ali will succeed him. When Ali departs, his son al-Hasan will succeed him, and when al-Hasan departs, al-Hujjah Muhammad al-Mahdi will succeed him. These are the twelve ones." That Jew, therefore, embraced Islam and praised Allah for having guided him."