= Solution
If two distinct <geodesic> arcs enclosed a disk, regard its boundary as a curvilinear two-sided polygon, with interior angles $\alpha,\beta$. These are positive; zero angle would make the arcs tangent and local <geodesic> uniqueness would make them coincide. The <geodesic>-curvature terms vanish, so Gauss-Bonnet gives
$$
\int_DK\,dA+(\pi-\alpha)+(\pi-\beta)=2\pi,\qquad\int_DK\,dA=\alpha+\beta>0.
$$
This contradicts $K\leq0$. Thus \b[no such disk-bounding <geodesic> bigon exists]. The disk hypothesis matters because it fixes the <Euler characteristic> on the right side.
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