# Way Below Relation is Auxiliary Relation

## Theorem

Lrt $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Then

$\ll$ is auxiliary relation

where $\ll$ denotes the way below relation.

## Proof

$\forall x, y \in S: x \ll y \implies x \preceq y$
$\forall x, y, z, u \in S: x \preceq y \ll z \preceq u \implies x \ll u$
$\forall x, y, z \in S: x \ll z \land y \ll z \implies x \vee y \ll z$
$\forall x: \bot \ll x$

where $\bot$ denotes the smallest element in $L$.

Thus by definition:

$\ll$ is auxiliary relation.

$\blacksquare$