Ali and Hassan leave Baba's service after Amir betrays Hassan, both by refusing to rescue him from rape and by trying to frame him for theft.
Amir cannot bear Hassan's presence in the house because it reminds him of his own guilt. When Hassan was being raped, Amir stood back and watched, too afraid to intervene on Hassan's behalf. Hassan is now a living reminder of that moment of weakness. Rather than make peace with that decision, Amir wants to avoid it. Due to his unwillingness to properly deal with painful feelings, Amir frames Hassan for theft to try to get Baba to dismiss him since he is otherwise compelled to keep both Ali and Hassan around.
The plan backfires. Hassan confesses to the theft when Baba confronts him about it, showing a willingness to sacrifice himself for Amir even when he is in the wrong. This only adds to Amir's feelings of guilt. Baba also forgives Hassan despite his distaste for thievery, but Hassan's father Ali insists on their leaving to live with a relative in Hazarajat. He holds Hassan protectively during this scene, implying that he knows what happened to the boy and wants to protect him from Amir. These protective feelings are at the heart of why Ali and Hassan depart.
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