Finer Supremum Precedes Supremum

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

Let $X, Y$ be subsets of $S$ such that
 * $X$ is finer than $Y$.

Then $\sup X \preceq \sup Y$

where $\sup X$ denotes the supremum of $X$.

Proof
We will prove that
 * $\sup Y$ is upper bound for $X$.

Let $x \in X$.

By definition of finer subset:
 * $\exists y \in Y: x \preceq y$

By definitions of supremum and upper bound:
 * $y \preceq \sup Y$

Thus by definition of transitivity:
 * $x \preceq \sup Y$

Hence by definition of supremum:
 * $\sup X \preceq \sup Y$