Not Preceding implies Approximating Relation and not Preceding

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

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

Let $\mathcal R$ be an approximating relation on $S$.

Then
 * $\exists u \in S: \left({u, x}\right) \in \mathcal R \land u \npreceq y$

Proof
By definition of approximating relation:
 * $x = \sup \left({x^{\mathcal R} }\right)$

By definition of supremum:
 * $y$ is upper bound for $x^{\mathcal R} \implies x \preceq y$

By definition of upper bound:
 * $\exists u \in S: u \in x^{\mathcal R} \land u \npreceq y$

Thus by definition of $\mathcal R$-segment:
 * $\exists u \in S: \left({u, x}\right) \in \mathcal R \land u \npreceq y$