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).
$\blacksquare$