# Strictly Succeed is Dual to Strictly Precede

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## Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $a, b \in S$.

The following are dual statements:

- $a$ strictly succeeds $b$
- $a$ strictly precedes $b$

## Proof

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

- $b \preceq a$ and $b \ne a$

The dual of this statement is:

- $a \succeq b$ and $b \ne a$

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

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

$\blacksquare$