Hartshorne II.6.5.b

Question

Let $X$ be a neotherian integral separated scheme wich is regular in codimension one. Let $Z$ be a proper closed subset of $X$ and $U=X\setminus Z$. In proposition II.6.5.b (page 133) Hartshorne says that if $\text{codim}(Z,X)\geqslant2$ then $\text{Cl}(X)\to\text{Cl}(U)$ is an isomorphism. Here $\text{Cl}(X)$ is the divisor (Weil) class group and the arrow take $Y$ prime divisor to $Y\cap U$. We know from part (a) that this arrow is surjective and we need injectivity that is for any divisor $D=\sum n_i Y_i$ if $\sum n_i Y_i\cap U=\text{div}(f)$ with $f\in K^*$ (function field of $X$ and $U$) then $D=\text{div}(g)$ with $g\in K^*$. I guess that $f=g$ hence I have to prove that for all prime divisor $Y$ (of $X$), $v_Y(f)=v_{Y\cap U}(f)$ ($v$ for valuation) hence I have to go to $\mathcal{O}_{\eta,X}$ and $\mathcal{O}_{\eta,U}$ hence locally $A_\mathfrak{p}$ and $(A_F)_\mathfrak{p}$. Are my reductions correct? Why valuations in this two local ring are the same? Thanks!