I have to prove that for a regular parametrized curve there is essentially (up to sign and a constant) a unique reparametrization which makes it a unit-speed curve. Let $x$ be a curve, $s(t) = \int_{t_0}^{t} \left \| \frac{\mathrm{d} x}{\mathrm{d} t} \right \| d\tau$ is an arc length parameter. We have the unit-speed reparametrization $y(s) = x(t(s))$, in fact
$1=\left \| \dot{y}(s) \right \| = \left \| \frac{\mathrm{d} x}{\mathrm{d} t} \right \| \, \frac{\mathrm{d} t}{\mathrm{d} s}.$
Suppose that u is a parameter that makes $y(u) = x(t(u))$ a unit-speed parametrization. Then
$1=\left \| \dot{y}(u) \right \| = \left \| \frac{\mathrm{d} x}{\mathrm{d} t} \right \| \, \frac{\mathrm{d} t}{\mathrm{d} u}$ and so $\frac{\mathrm{d} u}{\mathrm{d} t} = \pm \frac{\mathrm{d} s}{\mathrm{d} t} $ that finally yelds $u=\pm s + \mathrm{const}$. I have no idea if this can be proved as I did, can you spot any errors? Thanks