Ausmultiplizieren
1. Satz
\begin{align*}
(a_1 + a_2 + ... + a_n) (b_1 + b_2 + ... + b_k) =& \\
=& (\sum_{i=1}^n a_i) (\sum_{j=1}^k b_j) \\
=& \sum_{i=1}^n\sum_{j=1}^k a_ib_j
\end{align*}
2. Beweis
2.1. IA
\begin{align*}
(a_1) (b_1) =& a_1b_1 \\
=& \sum_{i=1}^1\sum_{j=1}^1 a_ib_j\\
(a_1) (b_1 + b_2) =& a_1b_1 + a_1b_2\\
=& \sum_{i=1}^1\sum_{j=1}^2 a_ib_j
\end{align*}
2.2. IA
\begin{align*}
(a_1 + ... + a_n)(b_1 + ... + b_{k} + b_{k+1}) =& \sum_{i=1}^n a_i)(\sum_{j=1}^{k} + b_{k+1}) \\
=& \sum_{i=1}^n\sum_{j=1}^k a_ib_j + \sum_{i=1}^na_ib_{k+1} \\
=& \sum_{i=1}^n\sum_{j=1}^{k+1}a_ib_j
\end{align*}