I'm trying to understand this proof in Fulton's algebraic book:
I understood why we can assume $C$ a closed subvariety of $\mathbb P^n$ such that $C\cap U_i\neq \emptyset$, $i=1,\ldots,n+1$ .
My problem is why this fact implies in the equality highlighted and why the author changes $X_i$ to $x_i$ (uppercase to lowercase).
I need help
Thanks in advance