Auxiliary Relation is Transitive

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

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

Then
 * $\mathcal R$ is transitive relation.

Proof
Let $x, y, z \in S$ such that
 * $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R$

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

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

Thus by condition $(ii)$ of auxiliary relation:
 * $\left({x, z}\right) \in \mathcal R$

Thus by definition
 * $\mathcal R$ is transitive relation.