## existence of such a one-to-one transformation

This question comes from the proof of Neyman's factorization theorem as excerpted below from Robert V. Hogg, Joseph W. McKean, Allen T. Craig, "Introduction to Mathematical Statistics", 6th edition:

In the proof, a one-to-one transformation is used which is indicated by the red line. But why does such a one-to-one transformation surely exists?

Although this theorem belongs to mathematical statistic, the question itself, I think, belongs to algebra, so I post it here.

Thank you for any help!