Connected Riemannian Manifold with Restricted Exponential Map defined on Whole Tangent Space admits Minimizing Geodesic Segment

Theorem
Let $\struct {M, g}$ be a connected Riemannian manifold.

Let $T_p M$ be the tangent space at $p \in M$.

Let $\exp_p$ be the restricted exponential map defined on the whole $T_p M$.

Then for all $q \in M$ there is a minimizing geodesic segment from $p$ to $q$.