Difference between "there exists at most one" and "there exists only one"

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What is the difference in proof of this type? I know that for "there exists only one" existence and uniqueness must be demonstrated but for "there exists at most one" I don't understand very well.

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"there exists only one" means "there exists (exactly) one". Whereas "there exists at most one" means either "there exists (exactly) one" or "there exists (exactly) none". So the statement, "there exists at most one integer between $0.5$ and $0.6$" is True, whereas the statement, "there exists only one integer between $0.5$ and $0.6$" is False.