Auxiliary Relation is Transitive

Theorem
Let $\struct {S, \vee, \preceq}$ be a bounded below join semilattice.

Let $\RR$ be relation on $S$ satisfying conditions $(i)$ and $(ii)$ of auxiliary relation.

Then
 * $\RR$ is a transitive relation.

Proof
Let $x, y, z \in S$ such that
 * $\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR$

By definition of reflexivity:
 * $z \preceq z$

By condition $(i)$ of auxiliary relation:
 * $x \preceq y$

Thus by condition $(ii)$ of auxiliary relation:
 * $\tuple {x, z} \in \RR$

Thus by definition
 * $\RR$ is a transitive relation.