\section{Summation of Zero/Indexed Summation}
Tags: Summations

\begin{theorem}
Let $a,b$ be [[Definition:Integer|integers]].
Let $\left[{a \,.\,.\, b}\right]$ denote the [[Definition:Integer Interval|integer interval]] between $a$ and $b$.
Let $0 : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be the [[Definition:Zero Mapping|zero mapping]].
Then the [[Definition:Indexed Summation|indexed summation]] of $0$ from $a$ to $b$ equals [[Definition:Zero of Standard Number System|zero]]:
:$\displaystyle \sum_{i \mathop = a}^b 0(i) = 0$
\end{theorem}

\begin{proof}
At least three proofs are possible:
* by induction, using [[Identity Element of Addition on Numbers]]
* using [[Indexed Summation of Multiple of Mapping]]
* using [[Indexed Summation of Sum of Mappings]]
{{ProofWanted}}
[[Category:Summations]]
3drcgu3jy45li9v3a1ivxrq7qiai7y4
\end{proof}


{"config": "wiki", "file": "thm_14835.txt"}