Non-Successor Element of Peano Structure is Unique

Theorem
Let $P$ be a set which fulfils the Peano Axiom schema:

Then:
 * $P \setminus s \left[{P}\right]$ is a singleton set

where:
 * $\setminus$ denotes set difference
 * $s \left[{P}\right]$ denotes the image of the mapping $s$.