Preceding is Top in Ordered Set of Auxiliary Relations

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

Let $\map {\it Aux} L$ be the set of all auxiliary relations on $S$.

Let $P = \struct {\map {\it Aux} L, \precsim}$ be an ordered set

where $\precsim \mathop = \subseteq \restriction_{\map {\it Aux} L \times \map {\it Aux} L}$

Then
 * $\preceq \mathop = \top_P$

where $\top_P$ denotes the greatest element in $P$.

Proof
By Preceding is Auxiliary Relation:
 * $\preceq \mathop \in \map {\it Aux} L$

By definition:
 * $\preceq$ is lower bound for $\O$ in $P$

We will prove that:
 * $\forall R \in \map {\it Aux} L: R$ is lower bound for $\O \implies R \mathop \precsim \preceq$

Let $R \in \map {\it Aux} L$

By condition $(i)$ of definition of auxiliary relation:
 * $R \mathop \subseteq \preceq$

Thus by definition of $\precsim$:
 * $R \mathop \precsim \preceq$

By definition of infimum:
 * $\preceq \mathop = \inf_P \O$

Thus by Infimum of Empty Set is Greatest Element:
 * $\preceq \mathop = \top_P$