proof explanation – Reverse function is enabled.

Continue on this question:

As you can see on this link

$$Phi = f ^ {- 1} circ h$$ is well defined and maps $$[c,d]$$ sure $$[a,b]$$ now in my textbook the next theorem following what i posted in the link, is the following:

Theorem 2
Let $$f:[a,b] subseteq R to R ^ n$$ and $$h:[c,d] subseteq R to R ^ n$$ to be 1-1 parametrizations of a simple arc C $$R ^ n$$ Then there is a unique function $$Phi$$ of [a,b] sure [c,d] such as $$h = f circ Phi$$. In addition, $$Phi$$ is continuous and strictly monotonous.

Now, I am confused because the proof provided for Theorem 2 is the one provided in the link ("Complementary Question") but in the supplementary question. $$Phi$$ maps [c,d] sure [a,b] and not in front. What do I miss?