In DoCarmo's Differential Geometry book there's an excerpt (on page 19-20) like this:

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Fundamental theorem of the local theory of curves:

Given differentiable functions $\displaystyle k(s) > 0$ and $\displaystyle \tau (s),$ where $\displaystyle s \in I$, there exists a regular parametrized curve $\displaystyle \alpha \colon I \rightarrow \mathrm{R^3}$

such that $\displaystyle s$ is the arc length, $\displaystyle k(s)$ is the curvature, and $\displaystyle \tau(s)$ is the torsion of $\displaystyle \alpha$. Moreover, any other curve $\displaystyle \bar{\alpha}$, satisfying the

same conditions, differs from $\displaystyle \alpha$ by a rigid motion; that is, there exists an orthogonal linear map $\displaystyle \rho$ of $\displaystyle \mathrm{R^3}$, with positive determinant,

and a vector $\displaystyle c$ such that $\displaystyle \bar{\alpha} = \rho \circ \alpha + c$.

.........

Proof of the Uniqueness Part of the Fundamental Theorem:

We first remark that arc length, curvature, and torsion are invariant under rigid motions; that means, for instance, that if $\displaystyle M \colon \mathrm{R^3} \rightarrow \mathrm{R^3}$

is a rigid motion and $\displaystyle \alpha = \alpha(t)$ is a parametrized curve, then

$\displaystyle \int_a^b \left|\frac{d{\alpha}}{dt}\right| dt = \int_a^b \left|\frac{d(M\circ \alpha)}{dt}\right| dt $

That is plausible, since these concepts are defined by using inner or vector products of certain derivatives (the derivatives are invariant under translations, and the

inner and vector products are expressed by means of lengths and angles of vectors, and thus also invariant under rigid motions).

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My question is:

why's that

$\displaystyle \int_a^b \left|\frac{d{\alpha}}{dt}\right| dt = \int_a^b \left|\frac{d(M\circ \alpha)}{dt}\right| dt $?

How did the author prove this? Is it possible to help me prove this? I'm completely lost at the last line of the excerpt given above.