What does $(A_1-\{a_1\},<_1\cap(A_1-\{a_1\})^2)$ mean in this context?

Question

I'm reading a proof of theorem 4.3 from textbook Introduction to set theory by Karel Hrbacek and Thomas Jech.

Here is the screenshot:

I have searched through the book from the beginning up to this theorem, but I have not found this notation before. Please explain what $(A_1-\{a_1\},<_1\cap(A_1-\{a_1\})^2)$ possibly means!

Answer 1

$\leq$ is a relation, and a relation on $A$ is a subset of $A^2$.

So, $(A_1-\{a_1\}, \leq_1\cap(A_1-\{a_1\})^2)$ is the set $(A_1-\{a_1\}$ with the relation $\leq_1\cap(A_1-\{a_1\})^2$. If we define $\leq_{1*}$ as $$\leq_{1*}: =\leq_1\cap(A_1-\{a_1\})^2$$

then $$x\leq_{1*}y\iff x\leq_1y$$ for all $x,y\in A_1$ so long as $x,y\neq a_1$. In short, $\leq_{1*}$ is simply the relation $\leq_1$, restricted to $A_1$.