# Proof

From the rules of finite cardinals and the union set operation it follows for any finite sets $$X$$ and $$U$$ with $$X\cap U=\emptyset$$

$|X\cup U|=|X|+|U|,~~~~~~~~~~~~~~~~~~~~( * )$

By hypothesis $$S$$ is a finite set, $$S_1,\ldots S_n\subset S$$ are its subsets forming a partition of $$S$$, i.e.

$S=\bigcup_{i=1}^n S_i,~~~~~~S_i\cap S_j=\emptyset,\forall i\neq j.$

and the cardinalities $$|S_i|$$ are all known $$i=1,\ldots n$$.

We observe that for $$1\le j\le n$$ we have $\left(\bigcup_{i=1}^{n-j}S_i\right)\cap \left(\bigcup_{i=n-j+1}^{n}S_i\right)=\emptyset.$ It follows from the rule $$( * )$$ that $\begin{array}{ccl} |S|=\left|\bigcup_{i=1}^n S_i\right|&=&\left|\bigcup_{i=1}^{n-1}S_i\right|+|S_n|\\ &=&\left|\bigcup_{i=1}^{n-2}S_i\right|+|S_{n-1}|+|S_n|\\ &\vdots&\\ &=&|S_1|+|S_2|+\ldots+|S_{n-1}|+|S_n|\\ &=&\sum_{i=1}^n |S_i| \end{array}$

Hereby we have used the associativity and commutativity of set union and the associativity and commutativity of the addition of natural numbers. .

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
2. Aigner, Martin: "Diskrete Mathematik", vieweg studium, 1993
3. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition