There is a lemma stating that there exists a unique vector field $G$ on the tangent bundle $TM$ of a manifold $M$ whose integral curves are of the form $t\mapsto (\gamma(t),\gamma’(t))$ where $\gamma$ is a geodesic.
The proof using a system of differential equations first shows that if we assume existence, then its uniqueness follows. However, to show existence the proof defines a vector field locally using the same system of differential equations but why must it define a smooth vector field?
The system of differential equations is $x_k’(t)=y_k, y_k’(t)=-\sum_{i,j}\Gamma_{ij}^k y_i y_j$, taking $(x_1,...,x_n,y_1,...,y_n)$ as the coordinate of a curve in $TM$