Flat connections have locally path-independent parallel transport on a coordinate ball

Shrink a chart and bundle trivialization about $p$ so that its coordinate image is a convex ball. Coordinatewise linear interpolation gives a fixed-endpoint homotopy $H(s,t)$ from ${\gamma }_0$ to ${\gamma }_1$; after a common finite subdivision it is smooth on each parameter rectangle. Let $V(s,t)$ be the [F2] parallel section along $t\mapsto H(s,t)$ with $V(s,0)=v$. In the fixed trivialization this is a linear ODE with smooth parameter $s$, so [F4] and uniqueness make $V$ smooth on each rectangle and continuous across the subdivision lines.

Put $W={\mathrm{\nabla }}_{{\mathrm{\partial }}_s}V$ along $H$. Expanding the two covariant derivatives in the fixed frame, using ${\mathrm{\nabla }}_{{\mathrm{\partial }}_t}V=0$ and $[{\mathrm{\partial }}_s,{\mathrm{\partial }}_t]=0$, gives ${\mathrm{\nabla }}_{{\mathrm{\partial }}_t}W=-R^{\mathrm{\nabla }}({\mathrm{\partial }}_{s}H,{\mathrm{\partial }}_{t}H)V=0$ by flatness. Because $H(s,0)=x$ and $V(s,0)=v$ are constant in $s$, $W(s,0)=0$; uniqueness in [F2] therefore gives $W=0$ on every rectangle and, successively, across all subdivision lines.

At $t=1$ the base point $H(s,1)=y$ is fixed, so $W(s,1)=0$ says that $s\mapsto V(s,1)\in E_y$ is constant. Hence $P_{{\gamma }_0}v=V(0,1)=V(1,1)=P_{{\gamma }_1}v$ by [F3]. This holds for every $v$, proving equality of the transport maps.