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Rika didn't defeat Takano becuase she knows about Rule XYZ. Rather, it's arguable that she knows about the rules, or at least the components of it, before, as hinted by her revelations to Akasaka in Himatsubushi-hen. However, as we can see, due to her stance and lack of effort, added with the strong will that drives Rule Y (which in turns feeds Rule X and Z), she was unable to do anything to prevent the tragerdy.
The key to victory in Matsuribayashi-hen is the miraculous disappearance of Rule X, enlisting and dissolving Rule Z, and preventing Rule Y from happening by knowing who the true culprit is before Rule Y happens.
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