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If Arachne had changed her attitude, I think the old woman would punish her in a very minimal manner and not turn her into a spider. I think there is no point punishing her if only she realized her mistakes and apologized about it but she remained proud and stubborn sticking to her belief that she owed no one of her talent/skill. Her pride and hard-headedness had lead her to a not so good situation. He was cursed and was turn into a spider which life's revolve only in spinning and weaving and nothing else. 
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