I need help with the proof of Theorem 4, Chapter I in E. Landau's "Foundations of Analysis"

Question

I need help with the proof of Theorem 4, Chapter I in E. Landau's "Foundations of Analysis"

To every pair of natural numbers $x$, $y$, we may assign in exactly one way a natural number, called $x+y$ such that:

1. $x+1=x'$ [where $x'$ has to be interpreted as the successor of $x$] and
2. $x+y'=(x+y)'$ for every $x$ and every $y$.

Now, he goes on:

First we will show that for each fixed $x$ there is at most one possibility of defining $x+y$ for all $y$ in such a way that: 1) and 2).

Here's the crucial passage I don't understand:

Let $a_y$ and $b_y$ be defined for all $y$ such that: $a_1$ and $b_1$ equal $x'$ and $a_{y'}$ equals $(a_y)'$ and $b_{y'}$ equals $(b_y)'$ for every $y$.

The problem to me is: who are $a_y$ and $b_y$ and either $a_1$ and $b_1$?

Answer 1

Landau wants to prove that there is only one binary function $+$ with the stated properties. This is equivalent to saying that for every fixed $x$, there is only one way to define a unary function that maps any number $y$ to a number $x+y$ and that satisfies the stated properties. (Note that given that $x$ is fixed, a function that maps every $y$ to some $x+y$ is unary and not binary.)

In order to prove this, Landau fixes $x$, considers two unary functions satisfying the stated properties, and then proves that they must be identical. In order to distinguish the two unary functions during the proof of their identity, he calls them $a_y$ and $b_y$.

The only statement that was correct in Federica's response is that $a_1$ and $b_1$ are simply $a_y$ and $b_y$ evaluated at $y=1$.

Answer 2

$a_1$ and $b_1$ are simply $a_y$ and $b_y$ evaluated at $y=1$. Since he's going to prove unicity of $y$ fixed $x$, such that $x+y$ satisfies $1)$ and $2)$, he supposes there are two such "y" and then proves they are actually the same. In order to distinguish one from the other, he calls them with $a$ and $b$. In order to recall they are both "candidates" to be "y", he puts a subscript $y$.

Answer 3

I am an Japanese almost 60 years old. I have been confused at the same place as you. I feel $a_y$ and $b_{y}$ are an expansion of $y_{1}$ and $y_{2}$. The $x$ is fixed but $y$ can be any natural number so $2$ natural number $y_{1}$ and $y_{2}$ shall be series. So author introduced $2$ series $a_y$ and $b_y$.

I thought your question is why $a_y$ and $b_{y}$ introduced on this definition and proof.