Doubts in Theorem 2.3.6-Differential geometry Andrew Pressley

I am reading Differential Geometry by Andrew Pressley(second edition) and have a doubt while reading Theorem 2.3.6 (Page 52). In the proof of this theorem there is a direct isometry M and I couldn't understand equation 2.18( in the proof). There it is given four equations and the last three equations except the first follows from Exercise 2.3.5. But I am not able to follow the first one, $\Gamma(s_0)=\tilde\gamma(s_0)$. I Will be extremely grateful if someone could help. Thanks in advance. Also if there is a geometric interpretation of the isometry M, the reading of the proof will be simpler for beginners.

$M$ first moves $\gamma(s_0)$ to $\tilde\gamma(s_0)$ (this is a translation), then rotates the Frenet frame of $\gamma$ at $s_0$ to coincide with the Frenet frame of $\tilde\gamma$ at $s_0$. Now the desired equation holds, by construction, at $s=s_0$. The interesting part of the proof is to show that, because of the Frenet equations, it holds for all $s$.