### Abstract

The theory's main theorem states that the cardinality of set partitions of
size k on a carrier set of size n is expressed by Stirling numbers of the
second kind. In Isabelle, Stirling numbers of the second kind are defined
in the AFP entry `Discrete Summation` through their well-known recurrence
relation. The main theorem relates them to the alternative definition as
cardinality of set partitions. The proof follows the simple and short
explanation in Richard P. Stanley's `Enumerative Combinatorics: Volume 1`
and Wikipedia, and unravels the full details and implicit reasoning steps
of these explanations.