Meet is Increasing

Theorem
Let $\struct {S, \preceq}$ be a meet semilattice.

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

Then:
 * $f$ is increasing as a mapping from the simple order product $\struct {S \times S, \precsim}$ of $\struct {S, \preceq}$ and $\struct {S, \preceq}$ into $\struct {S, \preceq}$.

Proof
Let $\tuple {x, y}, \tuple {z, t} \in S \times S$ such that:
 * $\tuple {x, y} \precsim \tuple {z, t}$

By definition of simple order product:
 * $x \preceq z$ and $y \preceq t$

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

By definition of $f$:
 * $\map f {x, y} \preceq \map f {z, t}$

Thus by definition:
 * $f$ is increasing mapping.