# Proof

(related to Proposition: Fundamental Counting Principle)

• From the rules of finite cardinals and the Cartesian product it follows for any finite sets $$X$$ and $$U$$ $|X\times U|=|X|\cdot |U|,~~~~~~~~~~~~~~~~~~~~( * )$
• By hypothesis $$S_1,\ldots S_n$$ are finite sets with the cardinalities $$|S_i|$$. It follows from the rule $$( * )$$ that $\begin{array}{ccl} |S|&=&|S_1\times\ldots\times S_{n-1}|\cdot|S_n|\\ &=&||S_1\times\ldots\times S_{n-2}|\cdot|S_{n-1}|\cdot|S_n|\\ &\vdots&\\ &=&|S_1|\cdot|S_2|\cdot\ldots\cdot|S_{n-1}|\cdot|S_n|\\ &=&\prod_{i=1}^n |S_i| \end{array}$
• Hereby, the associativity of Cartesian product and the associativity of the multiplication of natural numbers were used.

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