The following proposition is a generalization of the previous lemma (no. 1) .
Let \(S\) be a finite set and let its subsets \(S_1,\ldots S_n\subset S\) form a partition of \(S\), i.e.
\[S=\bigcup_{i=1}^n S_i,~~~~~~S_i\cap S_j=\emptyset,\forall i\neq j.\]
Further let the cardinalities \(|S_i|\) be known for all \(i=1,\ldots n\). Then the cardinality of the set \(S\) is given by the sum of the known cardinalities of its subsets.
\[|S|=\sum_{i=1}^n |S_i|.\]
Proofs: 1