Hartshorne Algebraic Geometry Proposition 1.5 (page5) In a notherian topological space $X$, every nomempty closed subset $Y$ can be expressed as a finite union $Y=Y_1 \cup \cdots \cup Y_r$ of irreducible closed subsets $Y_i$. If we require that $Y_i \nsupseteq Y_j$ for $i \neq j$, then the $Y_i$ are uniquely determined. They are called the irreducible components of $Y$.
I understood the existence of such a representation. But I can't understand a sentence in the uniqueness part, which I highlighted: Let $Y=Y_1 \cup \cdots \cup Y_r$ and $Y=Y'_1 \cup \cdots \cup Y'_s$ be two such representation. Then we can deduce that $Y_1=Y'_1$ (the details are in the textbook). Now let $Z=(Y-Y_1)^-$, then $Z=Y_2 \cup \cdots \cup Y_r$ and also $Z=Y'_2 \cup \cdots \cup Y'_s$.
What is the operator $^-$? I understood it as a closure, is it right? But then how can I deduce that $Z$ is represented as $Z=Y_2 \cup \cdots \cup Y_r=Y'_2 \cup \cdots \cup Y'_s$?