The question and its answer is given below:
But I could not understand 1- why he defined $\psi(\tilde{U}) = U$? I know that the importance of $\tilde{U}$ to adjust the dimension.
2-why $\psi ^{-1}= (x_{1}, x_{2}, ....., x_{k})$?
3- and why all of this leads to that $Z \cap U$ equal the required?
Could anyone explain this for me please?