=== $1$ implies $2$ === Let $\struct {S, \vee, \wedge, \preceq}$ adhere to Definition 1. Then since all finite non-empty suprema exist, we have: :$a \vee b = \sup \set {a, b} \in S$ for all $a, b \in S$, where $\vee$ denotes join. Hence $\struct {S, \vee, \preceq}$ is a join semilattice. Similarly, since all finite non-empty infima exist, we have: :$a \wedge b = \inf \set {a, b} \in S$ for all $a, b \in S$, where $\wedge$ denotes meet. Thus $\struct {S, \wedge, \preceq}$ is a meet semilattice. In conclusion, $\struct {S, \vee, \wedge, \preceq}$ satisfies Definition 2. {{qed|lemma}} === $2$ implies $1$ === Let $\struct {S, \vee, \wedge, \preceq}$ adhere to Definition 2. We will prove that: :$\struct {S, \vee, \preceq}$ is a join semilattice implies that: :$\struct {S, \preceq}$ admits all finite non-empty suprema From Supremum of Singleton, $\struct {S, \preceq}$ admits suprema of singletons. By definition of join, it also admits suprema of doubletons. Suppose now inductively that $\sup T \in S$ for all subsets $T$ of $S$ that have $n$ elements, where $n \ge 2$ is a natural number. Let now $V \subseteq S$ have $n + 1$ elements. Then $V = T \cup \set s$ for some $s \in S$ and $T \subseteq S$ with $n$ elements. By Supremum of Suprema, we have that: :$\sup V = \sup \set {\sup T, \sup \set s}$ By definition of join, this means $\sup V = \sup T \vee s$. Therefore, since $\struct {S, \vee, \preceq}$ is a join semilattice, $\sup V \in S$. By induction, $\sup T \in S$ for all finite non-empty $T \subseteq S$. That is, $S$ admits all finite non-empty suprema. By the Global Duality Principle and Dual Pairs (Order Theory), it follows that if: :$\struct {S, \wedge, \preceq}$ is a meet semilattice then it also true that: :$\struct {S, \preceq}$ admits all finite non-empty infima Hence $\struct {S, \vee, \wedge, \preceq}$ satisfies Definition 1. {{qed|lemma}} === $2$ implies $3$ === Suppose that $\struct {S, \vee, \wedge, \preceq}$ adheres to Definition 2. By virtue of Join Semilattice is Semilattice and Meet Semilattice is Semilattice, $\struct {S, \vee}$ and $\struct {S, \wedge}$ are semilattices. From Ordering Induced by Join Semilattice, the ordering mentioned in Definition 3 coincides with $\preceq$. The absorption laws stated are proved on Join Absorbs Meet and Meet Absorbs Join. Hence $\struct {S, \vee, \wedge, \preceq}$ adheres to Definition 3. {{qed|lemma}} === $3$ implies $2$ === Suppose that $\struct {S, \vee, \wedge, \preceq}$ adheres to Definition 3. Then $\preceq$ is thus defined by, for all $a, b \in S$: :$a \preceq b$ {{iff}} $a \vee b = b$ Suppose now that $a \vee b = b$. Then by absorption: :$a = a \wedge \paren {a \vee b} = a \wedge b$ Conversely, if $a \wedge b = a$, then by the other absorption law: :$a \vee b = \paren {a \wedge b} \vee b = b$ We conclude that also $a \preceq b$ {{iff}} $a \wedge b = a$. Next, to prove $a \vee b$ is a supremum for $\set {a, b}$. We have by absorption that $a = a \wedge \paren {a \vee b}$ so that $a \preceq a \vee b$, and similarly for $b$. Thus $a \vee b$ is an upper bound for $\set {a, b}$. Suppose that $c$ is an upper bound for $\set {a, b}$. Then: {{begin-eqn}} {{eqn | l = c       | r = a \vee c       | c = because $a \preceq c$ }} {{eqn | r = a \vee \paren {b \vee c}       | c = because $b \preceq c$ }} {{eqn | r = \paren {a \vee b} \vee c       | c = $\vee$ is associative }} {{end-eqn}} whence $a \vee b \preceq c$. Thus $a \vee b \in S$ is a supremum for $\set {a, b}$. Finally, we will show that $a \wedge b$ is an infimum for $\set {a, b}$. By Local Duality, $a \wedge b$ must be shown to be a supremum for $\set {a, b}$ with respect to the dual ordering $\succeq$ of $\preceq$. Since we have established that $a \succeq b$ {{iff}} $a \wedge b = b$, this directly follows from the above by interchanging $\wedge$ and $\vee$. Thus we have demonstrated $\struct {S, \vee, \wedge, \preceq}$ satisfies Definition 2. {{qed}}