(i) State and prove the Knaster-Tarski Fixed-Point Theorem.

(ii) A subset $S$ of a poset $X$ is called an up-set if whenever $x, y \in X$ satisfy $x \in S$ and $x \leqslant y$ then also $y \in S$. Show that the set of up-sets of $X$ (ordered by inclusion) is a complete poset.

Let $X$ and $Y$ be totally ordered sets, such that $X$ is isomorphic to an up-set in $Y$ and $Y$ is isomorphic to the complement of an up-set in $X$. Prove that $X$ is isomorphic to $Y$. Indicate clearly where in your argument you have made use of the fact that $X$ and $Y$ are total orders, rather than just partial orders.

[Recall that posets $X$ and $Y$ are called isomorphic if there exists a bijection $f$ from $X$ to $Y$ such that, for any $x, y \in X$, we have $f(x) \leqslant f(y)$ if and only if $x \leqslant y$.]