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I need help in order to fully understand the proof of Proposition 1.25 ... ...

Proposition 1.25 and its proof read as follows:

My questions are as follows:

**Question 1**

In the above text from Tapp we read the following:

" ... ... By the fundamental theorem of calculus, \(\displaystyle s'(t) = \mid \gamma ' (t) \mid \neq 0\), from which it can be seen that \(\displaystyle s\) is a smooth bijection onto \(\displaystyle \tilde{I}\) with nowhere-vanishing derivative. ... ... "

My question is as follows:

How do we know that \(\displaystyle s\) is a smooth bijection onto \(\displaystyle \tilde{I}\) ... how would we rigorously demonstrate this ... ... ?

**Question 2**In the above text from Tapp we read the following:

" ... ... Therefore, \(\displaystyle s\) has an inverse function \(\displaystyle \phi \ : \ \tilde{I} \to I\), which is also a smooth bijection with nowhere-vanishing derivative. ... ... "

My question is as follows:

How do we know that \(\displaystyle \phi\) is a smooth bijection with nowhere-vanishing derivative. ... ...

Presumably it is because the inverse of a smooth bijection is also a smooth bijection ... is that correct ...?

Hope someone can help ...

Peter