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
By Way Below implies Preceding:
 * $\forall x, y \in S: x \ll y \implies x \preceq y$

By Preceding and Way Below implies Way Below:
 * $\forall x, y, z, u \in S: x \preceq y \ll z \preceq u \implies x \ll u$

By Join is Way Below if Operands are Way Below:
 * $\forall x, y, z \in S: x \ll z \land y \ll z \implies x \vee y \ll z$

By Bottom is Way Below Any Element:
 * $\forall x: \bot \ll x$

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

Thus by definition:
 * $\ll$ is auxiliary relation.