First of all, would the prophet even appoint a leader to lead after him? Looking into history we see that in every city he appointed a governor and in every battle he appointed a commander. Then why would the prophet leave the leadership of the ummah an untouched subject. After being persecuted and all his effort in spreading Islam would he just leave the world without saying a word about the future. And in fact the prophet did advise the ummah on who to look for guidance after his death. And he did so on his last Hajj at a point known as Ghadeer.
On this day the Messenger of Allah spent approximately five ours in this place; three hours of which he was on the pulpit. He recited nearly one hundred verses from The Glorious Quran, and for seventy three times reminded and warned people of their deeds and future. Then he gave them along speech. The following is a part of his speech which has been widely narrated by the Sunni traditionists:
The Messenger of Allah declared: "It seems the time approached when I shall be called away (by Allah) and I shall answer that call. I am leaving for you two precious things and if you adhere both of them,you will never go astray after me. They are the Book of Allah and my progeny, that is my Ahlul-Bayt. The two shall never separate from each other until they come to me by the Pool (of Paradise)." (Sahih) Volume 6, Chapter 31, Hadith No. 3788
Then the Messenger of Allah continued: "Do I not have more right over
the believers than what they have over themselves?" People cried and
answered: "Yes, O' Messenger of God." Then Prophet held up the
hand of Ali and said: "Whoever I am his leader (Mawla), Ali is his
leader (Mawla). O' God, love those who love him, and be hostile to
those who are hostile to him." Sahih Tirmidhi, v2, p298, v5, p63
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