Multiplicative Auxiliary Relation iff Images are Filtered

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below lattice.

Let $\mathcal R$ be an auxiliary relation on $S$.

Then $\mathcal R$ is multiplicative
 * for all $x \in S$: $^{\mathcal R}x$ is filtered

where $^{\mathcal R}x$ denotes the $\mathcal R$-cosegment of $x$.

Sufficient Condition
Let $\mathcal R$ be multiplicative.

Let $x \in S$.

Let $a, b \in {}^{\mathcal R}x$.