Preceding is Auxiliary Relation

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Theorem

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


Then

$\preceq$ is auxiliary relation.


Proof

$\forall x, y \in S: x \preceq y \implies x \preceq y$

Then condition $(i)$ of auxiliary relation is satisfied.

By definition of transitivity:

$\forall x, y, z, u \in S: x \preceq y \preceq z \preceq u \implies x \preceq u$

Then the condition $(ii)$ of auxiliary relation is satisfied.

By definition of supremum:

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

Then the condition $(iii)$ of auxiliary relation is satisfied.

By definition of smallest element:

$\forall x \in S: \bot \preceq x$

Then the condition $(iv)$ of auxiliary relation is satisfied.

Thus the result by definition auxiliary relation.

$\blacksquare$


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