Not Preceding implies Approximating Relation and not Preceding

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Let $x, y \in S$ such that
 * $x \npreceq y$

Let $\RR$ be an approximating relation on $S$.

Then
 * $\exists u \in S: \tuple {u, x} \in \RR \land u \npreceq y$

Proof
By definition of approximating relation:
 * $x = \map \sup {x^\RR}$

By definition of supremum:
 * $y$ is upper bound for $x^\RR \implies x \preceq y$

By definition of upper bound:
 * $\exists u \in S: u \in x^\RR \land u \npreceq y$

Thus by definition of $\RR$-segment:
 * $\exists u \in S: \tuple {u, x} \in \RR \land u \npreceq y$