Auxiliary Relation is Congruent

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

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

Then
 * $\forall x, y, z, u \in S: \left({x, z}\right) \in \mathcal R \land \left({y, u}\right) \in \mathcal R \implies \left({x \vee y, z \vee u}\right) \in \mathcal R$

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

By definition of reflexivity:
 * $x \preceq x$ and $y \preceq y$

By Join Succeeds Operands:
 * $z \preceq z \vee u$ and $u \preceq z \vee u$

By condition $(ii)$ of auxiliary relation:
 * $\left({x, z \vee u}\right) \in \mathcal R$ and $\left({y, z \vee u}\right) \in \mathcal R$

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