Infimum Precedes Coarser Infimum

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

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

Then $\inf X \preceq \inf Y$

where $\inf X$ denotes the infimum of $X$.

Proof
We will prove that
 * $\inf X$ is lower bound for $Y$.

Let $x \in Y$.

By definition of coarser subset:
 * $\exists y \in X: y \preceq x$

By definitions of infimum and lower bound:
 * $\inf X \preceq y$

Thus by definition of transitivity:
 * $\inf X \preceq x$

Hence by definition of infimum:
 * $\inf X \preceq \inf Y$