Recall the theorem (first part) : It says that if X is a k-variety, then ![U_k(X)=\frac{k[X]^*}{k^*}](http://s0.wp.com/latex.php?latex=U_k%28X%29%3D%5Cfrac%7Bk%5BX%5D%5E%2A%7D%7Bk%5E%2A%7D&bg=f9fbf9&fg=666666&s=0 "U_k(X)=\frac{k[X]^*}{k^*}")
The first step is to be able to say “WLOG, we can assume X is a normal affine k-variety” .
For that, we should know what normalization is. Here’s a very well written post explaining what normalization actually is, with some worked out examples. And in case,the link becomes redundant at some point, a pdf copy of the same post is here : normalization
Now if Y is any open affine subset of our given X, we construct the normalization of Y, and call it 
![k[Y]\hookrightarrow k[Y ']](http://s0.wp.com/latex.php?latex=k%5BY%5D%5Chookrightarrow+k%5BY+%27%5D&bg=f9fbf9&fg=666666&s=0 "k[Y]\hookrightarrow k[Y ']")
![O_X(X)=k[X]\to O_X(Y)\cong k[Y]](http://s0.wp.com/latex.php?latex=O_X%28X%29%3Dk%5BX%5D%5Cto+O_X%28Y%29%5Ccong+k%5BY%5D&bg=f9fbf9&fg=666666&s=0 "O_X(X)=k[X]\to O_X(Y)\cong k[Y]")
So we have an injective map ![k[X]\hookrightarrow k[Y']](http://s0.wp.com/latex.php?latex=k%5BX%5D%5Chookrightarrow+k%5BY%27%5D&bg=f9fbf9&fg=666666&s=0 "k[X]\hookrightarrow k[Y']")
