How can one respond to someone who cites the Hadith, "Fatima is a part of me; whoever angers her has angered me," to prove her infallibility, based on the premise that her anger stems from the Prophet's anger, and the Prophet's anger stems from Allah's anger, thereby implying that she would not become angry over falsehood?

Refuting the شبهة (misconception/suspicion) of the Rafidah concerning Fatima’s anger, may Allah be pleased with her, and that the hadith (definition) related to it traces back to an incident involving Ali, may Allah be pleased with him, when he intended to propose to the daughter of Abu Jahl, which angered Fatima, may Allah be pleased with her, and Abu Bakr had no connection to that whatsoever. Furthermore, the Prophet’s anger, peace and blessings be upon him, was due to his concern for his daughter’s feelings, not her infallibility. Just as obedience to a ruler is conditional upon what is good and lawful, anger cannot be in opposition to the Sharia (definition) of Allah the Almighty, as evidenced by the Prophet’s saying, peace and blessings be upon him, "If Fatima, daughter of Muhammad, were to steal, I would cut off her hand."