Succeed is Dual to Precede

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

Let $a, b \in S$.

The following are dual statements:


 * $a$ succeeds $b$
 * $a$ precedes $b$

Proof
By definition, $a$ succeeds $b$ iff:


 * $b \preceq a$

The dual of this statement is:


 * $a \preceq b$

by Dual Pairs (Order Theory).

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

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

Also see

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