Functional composition and generalization of natural numbers by decomposition of semiring structure

In this paper we discover natural numbers in new algebraic object. The entity of the algebraic object which is seen as natural numbers is function, and from the functional composition we can have new generalization and analogs of natural numbers. From observation at the structure of them, we derive a combinatorial theorem which the addition and the multiplication of natural numbers satisfy, and we show that the theorem proves several important formulae in natural numbers.