I wanted to get some feedback on a proof on equivalence relations in the Book of Proof.
Suppose R is a reflexive and symmetric relation on a finite set A. Define a relation S on A by declaring xSy if and only if for some $n \in N$ there are elements $x_1, x_2, ... x_n \in A$ satisfying $xRx_1, x_1Rx_2, x_2Rx_3, ... ,x_{n-1}Rx_n$ and $x_nRy.$
Proposition 1. S is an equivalence relation.
Proof. First, we will prove S is reflexive. Suppose $x \in A$. Observe R is reflexive. So xRx. Thus, there is 1 element $x \in A$ satisfying xRx. Therefore S is reflexive.
Next, we will prove S is symmetric. Suppose $(a,b) \in S$. Then there is some $n \in N$ such that there are elements $x_1, x_2, x_3...x_n \in A$ satisfying $aRx_1, x_1Rx_2,... x_{n-1}Rx_n$, and $x_nRb$. Observe R is symmetric. So $bRx_n, x_nRx_{n-1}, x_{n-1}Rx_{n-2}... x_2Rx_1$, and $x_1Ra$. Thus $(b,a) \in S$. Therefore S is symmetric.
Now, we will prove S is transitive. Suppose $(a,b), (b,c) \in S$. Then, there exists $n, m \in N$ such that there are elements $x_1, x_2, x_3,... x_n \in A$ and $y_1, y_2, y_3,...y_m \in A$. So there exists a $w = n + m + 1$, such that there are elements $z_1, z_2, z_3, ... z_n,z_{n+1}...z_{w-1}, z_{w} \in A$ satisfying $aRz_1, z_2Rz_2,z_3Rz_4...z_{w-1}Rz_2, z_2Rc$, where $z_{n+1}=b, z_i=x_i$, and $z_j=y_{j-n-1}$ and $1 \le i \le n$ and $n+2 \le j \le n+m+1$. So $(a,c) \in S$. Therefore, S is transitive.
Proposition 2. The relation R is a subset of S.
Proof. Suppose $(a,b) \in R$. Observe that aRb, bRa, aRb. So aSb. Thus, $(a,b) \in S$. Therefore $R \subseteq S$.