Let $M$ be a smooth simply-connected manifold. Let $\nabla$ be a flat, symmetric connection on $M$. Let $p\in M$ and let $v,w\in T_pM$ belong to a normal neighborhood, such that the $\nabla$-geodesic triangle with vertices $p$, $\exp_p v$ and $\exp_p(v+w)$ is in $M$. I am looking for a proof that $$ \exp_p(v+w) = \exp_{\exp_p v}(\Pi_p^{\exp_p v} w), $$ where $\Pi_p^q$ denotes the (path-independent) parallel transport $T_pM\to T_qM$.

One can prove this by endowing $M$ with a metric, which is clearly Euclidean, and exploit the fact that this property holds trivially for Euclidean spaces (i.e., associativity of vector addition).

I am looking, however, for a direct proof that only uses the given properties of the connection.