How is the funeral prayer performed as authentically reported from the Prophet, may Allah bless him and grant him peace?

The description of the funeral prayer (Salat al-Janazah (definition)) is as follows: first, the pronouncement of the first Takbir (Allahu Akbar), then the recitation of Surat al-Fatiha and a short surah (definition). Then, the second Takbir and the recitation of blessings upon the Prophet (peace be upon him). Then, the third Takbir and the supplication for the deceased with the reported Du'a: "O Allah, forgive our living and our dead, our present and our absent, our young and our old, our males and our females. O Allah, whoever You keep alive among us, keep him alive upon Islam; and whoever You cause to die among us, cause him to die upon faith. O Allah, forgive him, have mercy on him, grant him well-being and pardon him. Make his arrival honorable and widen his entrance. Wash him with water, snow, and hail, and cleanse him of sins as a white garment is cleansed of dirt. O Allah, replace his home with a better home, and his family with a better family. O Allah, admit him to Paradise and protect him from the torment of the grave and the torment of the Fire. Make his grave spacious for him and illuminate it for him. O Allah, do not deprive us of his reward, and do not let us go astray after him." Then, the fourth Takbir, followed by a single Taslim (definition) to the right.