Bottom Relation is Bottom 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:
 * $\precsim \mathop = \subseteq \restriction_{ \map {\it Aux} L \times \map {\it Aux} L}$

Let $B = \set {\struct {\bot_L, x}: x \in S}$

where $\bot_L$ denotes smallest element in $L$.

Then
 * $B \mathop = \bot_P$

Proof
By Bottom Relation is Auxiliary Relation:
 * $B \in \map {\it Aux} L$

By definition:
 * $B$ is an upper bound for $\O$ in $P$

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

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

By condition $(iv)$ of definition of auxiliary relation:
 * $B \subseteq R$

Thus by definition of $\precsim$:
 * $B \precsim R$

By definition of supremum:
 * $B = \sup_P \O$

Thus by Supremum of Empty Set is Smallest Element:
 * $B = \bot_P$