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:
existence of such a one-to-one transformation-untitled.png

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!