I have a very simple question about quasiprojective varieties. I'm reading "Basic Algebraic Geometry" of Shafarevich.
Definition: A quasiprojective variety is an open subset of a closed projective set. A quasiprojective sub-variety $ Y \subset X$ is a subset $ Y \subset X$ such that $Y$ is itself a quasiprojective variety.
I want to prove that $ Y \subset X $ is a questiprojective subvariety if and only if $Y = Z - Z_a $ where $ Z,Z_a \subset X$ are closed projective sets.
Well obviously one side is direct. I have troubles with proving that if $ Y \subset X $ is a quasiprojective subvariety , then I can write $Y$ in that form.
I have the following : $ X = Z_1 \cap U_1 $ and $ Y = Z_2 \cap U_2 $ where $Z_i$ are closed projective sets and $U_i$ are open projective sets. And I also know that $ Y \subset X $ . Well... how can I write $Y$ in the form $Y = Z-Z_a $ where $Z, Z_a \subset X $ and closed projective sets? Please help me )=!