=== Sufficient Condition === Let $R$ be directed. We will prove by induction of the cardinality of finite subset of $H$. ==== Base case ==== :$\forall A \subseteq H: \left\vert{A}\right\vert = 0 \implies \exists h \in H: \forall a \in A: a \precsim h$ where $\left\vert{A}\right\vert$ denotes the cardinality of $A$. Let $A \subseteq H$ such that :$\left\vert{A}\right\vert = 0$ By Cardinality of Empty Set: :$A = \varnothing$ By definition of empty set: :$\exists h: h \in H$ Thus by definition of empty set: :$\exists h \in H: \forall a \in A: a \precsim h$ ==== Induction Hypothesis  ==== :$\forall A \subseteq H: \left\vert{A}\right\vert = n \implies \exists h \in H: \forall a \in A: a \precsim h$ ==== Induction Step ==== :$\forall A \subseteq H: \left\vert{A}\right\vert = n+1 \implies \exists h \in H: \forall a \in A: a \precsim h$ Let $A \subseteq H$ such that :$\left\vert{A}\right\vert = n+1$ By definition of cardinality of finite set: :$A \sim \N_{< n+1}$ where $\sim$ denotes set equivalence. By definition of set equivalence: :there exists a bijection $f: \N_{< n+1} \to A$ By Restriction of Injection is Injection: :$f \restriction_{\N_{< n}}: \N_{< n} \to f^\to\left({\N_{< n}}\right)$ is an injection. By definition :$f \restriction_{\N_{< n}}: \N_{< n} \to f^\to\left({\N_{< n}}\right)$ is a surjection. By definition :$f \restriction_{\N_{< n}}: \N_{< n} \to f^\to\left({\N_{< n}}\right)$ is a bijection. By definition of set equivalence: :$\N_{< n} \sim f^\to\left({\N_{< n}}\right)$ By definition of cardinality of finite set: :$\left\vert{f^\to\left({\N_{< n}}\right)}\right\vert = n$ By definitions of image of set and subset: :$f^\to\left({\N_{< n}}\right) \subseteq A$ By Subset Relation is Transitive: :$f^\to\left({\N_{< n}}\right) \subseteq H$ By Induction Hypothesis: :$\exists h \in H: \forall a \in f^\to\left({\N_{< n}}\right): a \precsim h$ By definition $\N_{< n+1}$ :$n \in \N_{< n+1}$ By definition of mapping: :$f \left({n}\right) \in A$ By definition of subset: :$f \left({n}\right) \in H$ By definition of directed subset: :$\exists h' \in H: f \left({n}\right) \precsim h' \land  h \precsim h'$ {{begin-eqn}} {{eqn |l = A       |r = f^\to\left({\N_{< n+1} }\right) | c = definition of surjection }} {{eqn |r =  f^\to\left({\N_{< n}\cup \left\{ {n}\right\} }\right) | c = definition of initial segment of natural numbers }} {{eqn |r = f^\to\left({\N_{< n} }\right) \cup f^\to\left({\left\{ {n}\right\} }\right) | c =Image of Union under Mapping }} {{eqn |r = f^\to\left({\N_{< n} }\right) \cup \left\{ {f\left({n }\right)}\right\} | c =Image of Singleton under Mapping }} {{end-eqn}} Let $a \in A$. Then by definitions of union and singleton: :$a \in f^\to\left({\N_{< n} }\right) \lor a = f\left({n }\right)$ So: :$a \precsim h \lor a = f\left({n }\right)$ Thus by definition of transitivity: :$a \precsim h'$ Thus: :$\exists h' \in H: \forall a \in A: a \precsim h'$ {{qed|lemma}} === Necessary Condition === Assume that :for every a finite subset $A$ of $H$ ::$\exists h \in H: \forall a \in A: a \precsim h$ Let $a, b \in H$ By definition of subset: :$\left\{ {a, b}\right\} \subseteq H$ :$\left\{ {a, b}\right\}$ is finite By assumption: :$\exists h \in H: \forall c \in \left\{ {a, b}\right\}: c \precsim h$ Thus by definition of unordered tuple: :$\exists h \in H: a \precsim h \land b \precsim h$ Thus by definition :$H$ is directed. {{qed}}