I found the solution at this website on page 8:

http://www.math.uregina.ca/~mareal/cs2.pdf

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]