I am trying to make sense of the following:
Theorem: Let $R$ be a relation on a set $A$. If $R$ is a well-ordered relation on $A$, then $R$ is a total-order relation on $A$.
Proof: Suppose $x,y\in A.$ Let $R=\{x,y\}\not=\emptyset$. By assumption, $R$ has a least element. If it is $x$, then $x\leq y$. If it is $y$, then $y\leq x$.
$$\text{__________________________________}$$
For a relation to be total ordered, then it must be that for each $x,y\in A$, $xRy$ and $yRx$. In this case we are comparing $x , y$ where one or the other is a least element. What about if neither is a least element though? Do we then simply say that $x,y$ must relate because if we consider whatever is the least element, say $z$, then both $x,y$ can relate to $z$ and therefore must relate to each other?
How could none of them be the last element? There is no other element in $\{x,y\}$ besides $x$ and $y$.