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Because their whole system depended upon the allies of Sparta
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being able to count on the Spartans to
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protect them from a third party, when it was necessary.
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So, if the Spartans said,
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well we're not going to do that,
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we'll leave it to some arbitrator to take care of,
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then they had to worry that the
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fundamental reason for the league,
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which gave them their power and their security,
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would disappear and that would be the end of that.
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They also had to worry that if they did not
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do what the Corinthians and the Megarians