Auxiliary Approximating Relation has Quasi Interpolation Property

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

Let $x, z \in S$.

Let $\mathcal R$ be an auxiliary approximating relation on $S$ such that
 * $\left({x, z}\right) \in \mathcal R \land x \ne z$

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

Proof
By definition of auxiliary relation:
 * $x \preceq z$

By definition of $\prec$:
 * $x \prec z$

By definition of antisymmetry:
 * $z \nprec x$

Then
 * $z \npreceq x$

By Not Preceding implies Approximating Relation and not Preceding:
 * $\exists u \in S: \left({u, z}\right) \land u \npreceq x$

Define $y = x \vee u$.

Thus by Join Succeeds Operands:
 * $x \preceq y$

Thus by definition of auxiliary relation:
 * $\left({y, z}\right) \in \mathcal R$

Thus by Preceding iff Join equals Larger Operand:
 * $x \ne y$