Strictly Succeed is Dual to Strictly Precede

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 $b$ :


 * $b \preceq a$ and $b \ne a$

The dual of this statement is:


 * $a \succeq b$ and $b \ne a$

by Dual Pairs (Order Theory).

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

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

Also see

 * Duality Principle (Order Theory)
 * Succeed is Dual to Precede