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Obviously, theorem 2.2 state the uniqueness of trajectory .but in the proof of lemma, why the uniqueness of trajectory means the uniqueness of $G$?

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    $\begingroup$ Notice that the value of $G$ at a given point $p$ is the derivative at $p$ of an integral curve of $G$ passing through $p$. This determines the vector field $G$. $\endgroup$ May 25, 2021 at 14:55

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Because the trajectory defines G. The condition placed on G is that it is defined by such a trajectory. Thus, there can only exist one such object: the one defined by the only trajectory that fulfills the required conditions.

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