= Algebraic partially ordered sets are continuous
Let $K_x$ be the <directed set> of <compact elements> below $x$, with <least upper bound> $x$, and put $A_x=\{y:y\ll x\}$. A <compact element> $k\leq x$ belongs to $A_x$: a <directed set> whose <least upper bound> is at least $x$ also has <least upper bound> at least $k$, so the definition of $k\ll k$ supplies an element above $k$. Hence $K_x\subseteq A_x$.
If $y_1,y_2\in A_x$, apply the <way-below relation> to $K_x$ to find $k_1,k_2\in K_x$ above them. Since $K_x$ is <directed>, it contains a common <upper bound> $k$ for $k_1,k_2$. This $k$ belongs to $A_x$, proving that $A_x$ is <directed>. Every member of $A_x$ is below $x$, while $K_x\subseteq A_x$ has <least upper bound> $x$. Therefore $x$ is the <least upper bound> of $A_x$, which is the defining property of a <continuous poset>.
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