# Upper Bound is Lower Bound for Inverse Ordering

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## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$.

Let $M$ be an upper bound for $\left({T, \preceq}\right)$.

Let $\succeq$ be the dual ordering of $\preceq$.

Then $M$ is a lower bound for $\left({T, \succeq}\right)$.

## Proof

Let $M$ be an upper bound for $\left({T, \preceq}\right)$.

That is:

- $\forall a \in T: a \preceq M$

By definition of dual ordering, it follows that:

- $\forall a \in T: M \succeq a$

That is, $M$ is a lower bound for $\left({T, \succeq}\right)$.

$\blacksquare$