A question about the lemma below
Lemma 2.3. There exists a unique vector field $G$ on $TM$ whose trajectories are of the form $t \to (\gamma(t),\gamma'(t))$ where $\gamma$ is a geodesic on $M$.
Proof:
We shall first prove the uniquiness of $G$, supposing its existence. Consider a system of coordinates $(U,x)$ on $M$. From the hypothesis, the trajectories of $G$ on $TU$ are given by $t \to (\gamma(t),\gamma'(t))$ where $\gamma$ is a geodesic. It follows that $t \to (\gamma(t),\gamma'(t))$ is a solution of the system of differential equations (1'). From the uniqueness of the trajectories of such system, we conclude that if $G$ exists, then it is unique. To prove the existence of $G$, define it locally by the system (1'). Using the uniqueness, we conclude that $G$ is well-defined on $TM$.
If you need I can write the system of ODE, but I don't think it's relevant.
What it seems to me is that the author from one side proves the uniqueness assuming existence, and later it defines the $G$ by the mentioned system of ODE and from there we know that the solution is unique.
Couldn't the author just skip the first part and move on with the definition using the system of ODE?