# Succeed is Dual to Precede

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## Theorem

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

Let $a, b \in S$.

The following are dual statements:

## Proof

By definition, $a$ succeeds $b$ if and only if:

- $b \preceq a$

The dual of this statement is:

- $a \preceq b$

By definition, this means $a$ precedes $b$.

The converse follows from Dual of Dual Statement (Order Theory).

$\blacksquare$