How would we show that the knowledge of the vector function of the normal vector of a curve a, determines the curvature and the torsion of a. Assuming that the curve has nonzero torsion everywhere ?
If you have an expression for the normal vector, the curvature and torsion are there in the Frenet equation (differentiate the normal vector). Not sure that really answers your question but hope it does.
I don't know if other approaches exist to deriving the Frenet equations, but in my text the curvature is simply defined as the factor that makes unit length, and after that the author mentions the geometric interpretation. What I'm saying is that in my text (as well as in a supporting text, actually) we don't have curvature as a geometric "thing" and then discover that the curvature pops out in such-and-such a computation, rather curvature is defined as what popped out and then it's noted that in a particular way this helps describe the curve. My text handles deriving torsion the same way--it plops out of a calculation, and the author notes "since we can't immediately identify it in terms of known quantities, we give it a name". Thus, torsion is defined and, again, not something discovered requiring proof. I dunno, my reasoning in this domain feels impaired but this still seems funky.