Admissible Curve in Riemannian Manifold has Unique Forward Reparametrization by Arc Length

Theorem
Let $\struct {M, g}$ be a Riemannian manifold with or without boundary.

Let $I \subseteq \R$ be a real interval.

Let $\gamma : I \to M$ be an admissible Curve.

Then $\gamma$ has a unique forward reparametrization by arc length.