Meet is Increasing

Theorem
Let $\left({S, \preceq}\right)$ be a meet semilattice.

Let $f: S \times S \to S$ be a mapping such that
 * $\forall s, t \in S: f\left({s, t}\right) = s \wedge t$

Then:
 * $f$ is increasing as a mapping from Cartesian product $\left({S\times S, \precsim}\right)$ of $\left({S, \preceq}\right)$ and $\left({S, \preceq}\right)$ into $\left({S, \preceq}\right)$.

Proof
Let $\left({x, y}\right)$, $\left({z, t}\right) \in S \times S$ such that
 * $\left({x, y}\right) \precsim \left({z, t}\right)$

By definition of Cartesian product of ordered sets:
 * $x \preceq z$ and $y \preceq t$

By Meet Semilattice is Ordered Structure:
 * $x \wedge y \preceq z \wedge t$

By definition of $f$:
 * $f\left({x, y}\right) \preceq f\left({z, t}\right)$

Thus by definition
 * $f$ is increasing mapping.