I found the solution at this website on page 8:
The proof is something like this:
To prove the uniqueness part, we choose the linear orthogonal transformation which maps the Frenet fame of at to the Frenet frame of at .
Note that both frames are orthonormal systems of vectors with the last vector equal to the vector product of the previous two: thus the transformation mentioned above exists and moreover, has
We have that , has unique solution for all , by using a uniqueness property of the solutions of the system.[Q.E.D]