From baef622fc53b1842cbc21cdb950f7694f61aee2a Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Wed, 18 Oct 2023 21:33:51 +0200
Subject: [PATCH] main > main: musterlsg

---
 docs/musterloesung_WiSe_2020_2021_Kl2_A5.pdf |  Bin 0 -> 133502 bytes
 docs/musterloesung_WiSe_2020_2021_Kl2_A5.tex | 1655 ++++++++++++++++++
 2 files changed, 1655 insertions(+)
 create mode 100644 docs/musterloesung_WiSe_2020_2021_Kl2_A5.pdf
 create mode 100644 docs/musterloesung_WiSe_2020_2021_Kl2_A5.tex

diff --git a/docs/musterloesung_WiSe_2020_2021_Kl2_A5.pdf b/docs/musterloesung_WiSe_2020_2021_Kl2_A5.pdf
new file mode 100644
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diff --git a/docs/musterloesung_WiSe_2020_2021_Kl2_A5.tex b/docs/musterloesung_WiSe_2020_2021_Kl2_A5.tex
new file mode 100644
index 0000000..47808ad
--- /dev/null
+++ b/docs/musterloesung_WiSe_2020_2021_Kl2_A5.tex
@@ -0,0 +1,1655 @@
+%% ********************************************************************************
+%% AUTHOR:     Raj Dahya
+%% CREATED:    26. März 2020
+%% EDITED:     März-April 2020
+%% TYPE:       Notizen
+%% TITLE:      Musterlösung Klausur2 WiSe 2020/2021, A5
+%% DOI:        —
+%% DEPARTMENT: Fakultät for Mathematik und Informatik
+%% INSTITUTE:  Universität Leipzig
+%% ********************************************************************************
+
+%% ********************************************************************************
+%% DOCUMENT STRUCTURE:
+%% ~~~~~~~~~~~~~~~~~~~
+%%
+%%   - root.tex;
+%%    |
+%%    ---- parameters.tex;
+%%    |
+%%    ---- srclocal/index.tex;
+%%        |
+%%        ---- src/setup-type.tex;
+%%        |
+%%        ---- src/setup-packages.tex;
+%%        |
+%%        ---- src/setup-parameters.tex;
+%%        |
+%%        ---- src/setup-macros.tex;
+%%        |
+%%        ---- src/setup-environments.tex;
+%%        |
+%%        ---- src/setup-layout.tex;
+%%        |
+%%        ---- srclocal/setup-locallayout.tex;
+%%        |
+%%        ---- srclocal/setup-localmacros.tex;
+%%    |
+%%    ---- body/index.tex;
+%%        |
+%%        ---- body/A5.tex;
+%%            |
+%%            ---- body/A5a.tex;
+%%            |
+%%            ---- body/A5b.tex;
+%%
+%% DOCUMENT-RANDOM-SEED: 5637845
+%% ********************************************************************************
+
+%% ********************************************************************************
+%% FILE: root.tex
+%% ********************************************************************************
+
+%% ********************************************************************************
+%% FILE: parameters.tex
+%% ********************************************************************************
+
+%% ********************************************************************************
+%% FILE: srclocal/index.tex
+%% ********************************************************************************
+
+\makeatletter
+
+%% ********************************************************************************
+%% FILE: src/setup-type.tex
+%% ********************************************************************************
+
+\documentclass[
+    10pt,
+    a4paper,
+    oneside,
+    openright,
+    center,
+    chapterbib,
+    crosshair,
+    fleqn,
+    headcount,
+    headline,
+    indent,
+    indentfirst=false,
+    portrait,
+    phonetic,
+    oldernstyle,
+    onecolumn,
+    sfbold,
+    upper,
+]{scrbook}
+
+%% ********************************************************************************
+%% FILE: src/setup-packages.tex
+%% ********************************************************************************
+
+\PassOptionsToPackage{T2A,OT1}{fontenc}   % T1,OT1,T2A,OT2
+\PassOptionsToPackage{utf8}{inputenc} % utf8
+\PassOptionsToPackage{british,english,ngerman,russian}{babel}
+\PassOptionsToPackage{
+    english,
+    ngerman,
+    russian,
+    capitalise,
+}{cleveref}
+\PassOptionsToPackage{
+    bookmarks=true,
+    bookmarksopen=false,
+    bookmarksopenlevel=0,
+    bookmarkstype=toc,
+    colorlinks=false,
+    raiselinks=true,
+    hyperfigures=true,
+}{hyperref}
+\PassOptionsToPackage{
+    reset,
+    left=1in,
+    right=1in,
+    top=20mm,
+    bottom=20mm,
+    heightrounded,
+}{geometry}
+\PassOptionsToPackage{
+    framemethod=TikZ,
+}{mdframed}
+\PassOptionsToPackage{normalem}{ulem}
+\PassOptionsToPackage{
+    amsmath,
+    thmmarks,
+}{ntheorem}
+\PassOptionsToPackage{table}{xcolor}
+\PassOptionsToPackage{
+    all,
+    color,
+    curve,
+    frame,
+    import,
+    knot,
+    line,
+    movie,
+    rotate,
+    textures,
+    tile,
+    tips,
+    web,
+    xdvi,
+}{xy}
+
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{ntheorem} % <— muss nach den ams* Packages vorkommen!!
+\usepackage{array}
+\usepackage{babel}
+\usepackage{bbding}
+\usepackage{bbm}
+\usepackage{calc}
+\usepackage{sectsty}
+\usepackage{titlesec}
+\usepackage{fancyhdr}
+\usepackage{footmisc}
+\usepackage{geometry}
+\usepackage{graphicx}
+\usepackage{ifpdf}
+\usepackage{ifthen}
+\usepackage{ifnextok}
+\usepackage{longtable}
+\usepackage{multicol}
+\usepackage{multirow}
+\usepackage{nameref}
+\usepackage{nowtoaux}
+\usepackage{paralist}
+\usepackage{enumerate} %% nach [paralist]
+\usepackage{pgf}
+\usepackage{arydshln} %% nach [pgf]!
+\usepackage{pgfplots}
+\usepackage{proof}
+\usepackage{refcount}
+\usepackage{relsize}
+\usepackage{savesym}
+\usepackage{stmaryrd}
+\usepackage{subfigure}
+\usepackage{yfonts}   %% <— Altgotische Fonts
+\usepackage{tikz}
+\usepackage{xy}
+\usepackage{undertilde}
+\usepackage{ulem} %% <– f\"ur besseren \underline-Befehl (\ul)
+\usepackage{xcolor}
+\usepackage{xspace}
+\usepackage{xstring}
+\usepackage{hyperref}
+\usepackage{cleveref} % must vor hyperref geladen werden.
+
+\pgfplotsset{compat=newest}
+
+\usetikzlibrary{
+    angles,
+    arrows,
+    automata,
+    calc,
+    decorations,
+    decorations.pathmorphing,
+    decorations.pathreplacing,
+    math,
+    positioning,
+    patterns,
+    quotes,
+    snakes,
+}
+
+%% \var ≈ alter Befehl
+%% \xvar ≈ wie das neue Package \var interpretieren soll.
+\savesymbol{Diamond}
+\savesymbol{emptyset}
+\savesymbol{ggg}
+\savesymbol{int}
+\savesymbol{lll}
+\savesymbol{RectangleBold}
+\savesymbol{langle}
+\savesymbol{rangle}
+\savesymbol{hookrightarrow}
+\savesymbol{hookleftarrow}
+\savesymbol{Asterisk}
+\usepackage{mathabx}
+\usepackage{wasysym}
+\let\varemptyset=\emptyset
+\restoresymbol{x}{Diamond}
+\restoresymbol{x}{emptyset}
+\restoresymbol{x}{ggg}
+\restoresymbol{x}{int}
+\restoresymbol{x}{lll}
+\restoresymbol{x}{RectangleBold}
+\restoresymbol{x}{langle}
+\restoresymbol{x}{rangle}
+\restoresymbol{x}{hookrightarrow}
+\restoresymbol{x}{hookleftarrow}
+\restoresymbol{x}{Asterisk}
+
+\ifpdf
+    \usepackage{pdfcolmk}
+\fi
+
+\usepackage{mdframed}
+
+%% Force-Import aus MnSymbol
+\DeclareFontFamily{U}{MnSymbolA}{}
+\DeclareFontShape{U}{MnSymbolA}{m}{n}{
+    <-6> MnSymbolA5
+    <6-7> MnSymbolA6
+    <7-8> MnSymbolA7
+    <8-9> MnSymbolA8
+    <9-10> MnSymbolA9
+    <10-12> MnSymbolA10
+    <12-> MnSymbolA12
+}{}
+\DeclareFontShape{U}{MnSymbolA}{b}{n}{
+    <-6> MnSymbolA-Bold5
+    <6-7> MnSymbolA-Bold6
+    <7-8> MnSymbolA-Bold7
+    <8-9> MnSymbolA-Bold8
+    <9-10> MnSymbolA-Bold9
+    <10-12> MnSymbolA-Bold10
+    <12-> MnSymbolA-Bold12
+}{}
+\DeclareSymbolFont{MnSyA}{U}{MnSymbolA}{m}{n}
+\DeclareMathSymbol{\lcirclearrowright}{\mathrel}{MnSyA}{252}
+\DeclareMathSymbol{\lcirclearrowdown}{\mathrel}{MnSyA}{255}
+\DeclareMathSymbol{\rcirclearrowleft}{\mathrel}{MnSyA}{250}
+\DeclareMathSymbol{\rcirclearrowdown}{\mathrel}{MnSyA}{251}
+
+\DeclareFontFamily{U}{MnSymbolC}{}
+\DeclareSymbolFont{MnSyC}{U}{MnSymbolC}{m}{n}
+\DeclareFontShape{U}{MnSymbolC}{m}{n}{
+    <-6>  MnSymbolC5
+    <6-7>  MnSymbolC6
+    <7-8>  MnSymbolC7
+    <8-9>  MnSymbolC8
+    <9-10> MnSymbolC9
+    <10-12> MnSymbolC10
+    <12->   MnSymbolC12%
+}{}
+\DeclareMathSymbol{\powerset}{\mathord}{MnSyC}{180}
+
+%% ********************************************************************************
+%% FILE: src/setup-parameters.tex
+%% ********************************************************************************
+
+\def\boolwahr{true}
+\def\boolfalsch{false}
+\def\boolleer{}
+
+\let\documenttwosided\boolfalsch
+\let\boolinappendix\boolfalsch
+\let\boolinmdframed\boolfalsch
+\let\eqtagset\boolfalsch
+\let\eqtaglabel\boolleer
+\let\eqtagsymb\boolleer
+
+\newcount\bufferctr
+\newcount\bufferreplace
+\newcounter{columnanzahl}
+
+\newlength\rtab
+\newlength\gesamtlinkerRand
+\newlength\gesamtrechterRand
+\newlength\ownspaceabovethm
+\newlength\ownspacebelowthm
+\setlength{\rtab}{0.025\textwidth}
+\setlength{\ownspaceabovethm}{0.5\baselineskip}
+\setlength{\ownspacebelowthm}{0.5\baselineskip}
+\setlength{\gesamtlinkerRand}{0pt}
+\setlength{\gesamtrechterRand}{0pt}
+
+\def\secnumberingpt{$\cdot$}
+\def\secnumberingseppt{.}
+\def\subsecnumberingseppt{}
+\def\thmnumberingpt{$\cdot$}
+\def\thmnumberingseppt{}
+\def\thmForceSepPt{.}
+
+\definecolor{leer}{gray}{1}
+\definecolor{hellgrau}{gray}{0.85}
+\definecolor{dunkelgrau}{gray}{0.5}
+\definecolor{maroon}{rgb}{0.6901961,0.1882353,0.3764706}
+\definecolor{dunkelgruen}{rgb}{0.015625,0.363281,0.109375}
+\definecolor{dunkelrot}{rgb}{0.5450980392,0,0}
+\definecolor{dunkelblau}{rgb}{0,0,0.5450980392}
+\definecolor{blau}{rgb}{0,0,1}
+\definecolor{newresult}{rgb}{0.6,0.6,0.6}
+\definecolor{improvedresult}{rgb}{0.9,0.9,0.9}
+\definecolor{hervorheben}{rgb}{0,0.9,0.7}
+\definecolor{starkesblau}{rgb}{0.1019607843,0.3176470588,0.8156862745}
+\definecolor{achtung}{rgb}{1,0.5,0.5}
+\definecolor{frage}{rgb}{0.5,1,0.5}
+\definecolor{schreibweise}{rgb}{0,0.7,0.9}
+\definecolor{axiom}{rgb}{0,0.3,0.3}
+
+%% ********************************************************************************
+%% FILE: src/setup-macros.tex
+%% ********************************************************************************
+
+%% ****************************************************************
+%% TEX:
+%% ****************************************************************
+
+\def\let@name#1#2{\expandafter\let\csname #1\expandafter\endcsname\csname #2\endcsname\relax}
+\DeclareRobustCommand\crfamily{\fontfamily{ccr}\selectfont}
+\DeclareTextFontCommand{\textcr}{\crfamily}
+
+\def\nichtzeigen#1{\phantom{#1}}
+
+%% ****************************************************************
+%% SPACING:
+%% ****************************************************************
+
+\def\ifthenelseleer#1#2#3{\ifthenelse{\equal{#1}{}}{#2}{#1#3}}
+\def\bedingtesspaceexpand#1#2#3{\ifthenelseleer{\csname #1\endcsname}{#3}{#2#3}}
+\def\voritemise{\leavevmode\nvraum{1}}
+\def\hraum{\null\hfill\null}
+\def\vraum{\null\vfill\null}
+\def\nvraum{\@ifnextchar\bgroup{\nvraum@c}{\nvraum@bes}}
+    \def\nvraum@c#1{\vspace*{-#1\baselineskip}}
+    \def\nvraum@bes{\vspace*{-\baselineskip}}
+\def\erlaubeplatz{\relax\ifmmode\else\@\xspace\fi}
+\def\entferneplatz{\relax\ifmmode\else\expandafter\@gobble\fi}
+\def\forceindent{\hspace*{20pt}} %% * nötig, damit am Anfang/Ende einer Zeile nicht ignoriert wird
+
+%% ****************************************************************
+%% TAGS / BEZEICHNUNGEN / LABELLING:
+%% ****************************************************************
+
+\def\send@toaux#1{\@bsphack\protected@write\@auxout{}{\string#1}\@esphack}
+
+%% \rlabel{LABEL}[CTR]{CREF-SHORT}{CREF-LONG}{DISPLAYTEXT}
+\def\rlabel#1[#2]#3#4#5{#5\rlabel@aux{#1}[#2]{#3}{#4}{#5}}
+    \def\rlabel@aux#1[#2]#3#4#5{%
+        \send@toaux{\newlabel{#1}{{\@currentlabel}{\thepage}{{\unexpanded{#5}}}{#2.\csname the#2\endcsname}{}}}\relax%
+    }
+
+%% \tag@rawscheme{CREF-SHORT}{CREF-LONG}[CTR]{LEFT-BRKT}{RIGHT-BRKT}    [LABEL]{DISPLAYTEXT}
+\def\tag@rawscheme#1#2[#3]#4#5{\@ifnextchar[{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}}{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}[*]}}
+    \def\tag@rawscheme@#1#2[#3]#4#5[#6]{\@ifnextchar\bgroup{\tag@rawscheme@@{#1}{#2}[#3]{#4}{#5}[#6]}{\tag@rawscheme@@{#1}{#2}[#3]{#4}{#5}[#6]{}}}
+    \def\tag@rawscheme@@#1#2[#3]#4#5[#6]#7{%
+        \ifthenelse{\equal{#6}{*}}{%
+            \ifthenelse{\equal{#7}{\boolleer}}{\refstepcounter{#3}#4\csname the#3\endcsname#5}{#4#7#5}%
+        }{%
+            \refstepcounter{#3}#4%
+            \ifthenelse{\equal{#7}{\boolleer}}{\rlabel{#6}[#3]{#1}{#2}{\csname the#3\endcsname}}{\rlabel{#6}[#3]{#1}{#2}{#7}}%
+            #5%
+        }%
+    }
+%% \tag@scheme{CREF-SHORT}{CREF-LONG}[CTR]    [LABEL]{DISPLAYTEXT}
+\def\tag@scheme#1#2[#3]{\tag@rawscheme{#1}{#2}[#3]{\upshape(}{\upshape)}}
+
+%% \eqtag[LABEL]{DISPLAYTEXT}
+\def\eqtag@post#1{\makebox[0pt][r]{#1}}
+\def\eqtag@pre{\tag@scheme{Eq}{Equation}[Xe]}
+\def\eqtag{\@ifnextchar[{\eqtag@}{\eqtag@[*]}}
+    \def\eqtag@[#1]{\@ifnextchar\bgroup{\eqtag@@[#1]}{\eqtag@@[#1]{}}}
+    \def\eqtag@@[#1]#2{\eqtag@post{\eqtag@pre[#1]{#2}}}
+
+\def\eqcref#1{\text{(\ref{#1})}}
+\def\ptcref#1{\ref{#1}}
+\def\punktlabel#1{\label{it:#1:\beweislabel}}
+\def\punktcref#1{\eqcref{it:#1:\beweislabel}}
+\def\crefit#1#2{\cref{#1}~\eqcref{it:#2:#1}}
+\def\Crefit#1#2{\Cref{#1}~\eqcref{it:#2:#1}}
+
+%% UNDER/OVERSET BEFEHLE
+\def\opfromto[#1]_#2^#3{\underset{#2}{\overset{#3}{#1}}}
+\def\textoverset#1#2{\overset{\text{#1}}{#2}}
+\def\textunderset#1#2{\underset{#2}{\text{#1}}}
+\def\crefoverset#1#2{\textoverset{\cref{#1}}{#2}}
+\def\Crefoverset#1#2{\textoverset{\Cref{#1}}{#2}}
+\def\crefunderset#1#2{\textunderset{#2}{\cref{#1}}}
+\def\Crefunderset#1#2{\textunderset{#2}{\Cref{#1}}}
+\def\eqcrefoverset#1#2{\textoverset{\eqcref{#1}}{#2}}
+\def\eqcrefunderset#1#2{\textunderset{#2}{\eqcref{#1}}}
+\def\mathclap#1{#1}
+\def\oberunterset#1{\@ifnextchar^{\oberunterset@oben{#1}}{\oberunterset@unten{#1}}}
+    \def\oberunterset@oben#1^#2_#3{\underset{\mathclap{#3}}{\overset{\mathclap{#2}}{#1}}}
+    \def\oberunterset@unten#1_#2^#3{\underset{\mathclap{#2}}{\overset{\mathclap{#3}}{#1}}}
+    \def\breitunderbrace#1_#2{\underbrace{#1}_{\mathclap{#2}}}
+    \def\breitoverbrace#1^#2{\overbrace{#1}^{\mathclap{#2}}}
+    \def\breitunderbracket#1_#2{\underbracket{#1}_{\mathclap{#2}}}
+    \def\breitoverbracket#1^#2{\overbracket{#1}^{\mathclap{#2}}}
+
+\def\generatenestedsecnumbering#1#2#3{%
+    \expandafter\gdef\csname thelong#3\endcsname{%
+        \expandafter\csname the#2\endcsname%
+        \secnumberingpt%
+        \expandafter\csname #1\endcsname{#3}%
+    }%
+    \expandafter\gdef\csname theshort#3\endcsname{%
+        \expandafter\csname #1\endcsname{#3}%
+    }%
+}
+\def\generatenestedthmnumbering#1#2#3{%
+    \expandafter\gdef\csname the#3\endcsname{%
+        \expandafter\csname the#2\endcsname%
+        \thmnumberingpt%
+        \expandafter\csname #1\endcsname{#3}%
+    }%
+    \expandafter\gdef\csname theshort#3\endcsname{%
+        \expandafter\csname #1\endcsname{#3}%
+    }%
+}
+
+%% ****************************************************************
+%% ALLG. MACROS:
+%% ****************************************************************
+
+\def\+#1{\addtocounter{#1}{1}}
+\def\setcounternach#1#2{\setcounter{#1}{#2}\addtocounter{#1}{-1}}
+\def\textsubscript#1{${}_{\textup{#1}}$}
+\def\rome#1{\overline{\underline{#1}}}
+\def\textTODO{\text{[{\large\textcolor{red}{More work needed!}}]}}
+\def\hlineEIGENpt{\hdashline[0.5pt/5pt]}
+\def\clineEIGENpt#1{\cdashline{#1}[0.5pt/5pt]}
+
+\def\forcepunkt#1{#1\IfEndWith{#1}{.}{}{.}}
+\def\lateinabkuerzung#1#2{%
+    \expandafter\gdef\csname #1\endcsname{\emph{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}}
+}
+\def\deutscheabkuerzung#1#2{%
+    \expandafter\gdef\csname #1\endcsname{{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}}
+}
+
+%% ****************************************************************
+%% MATHE
+%% ****************************************************************
+
+\def\matrix#1{\left(\begin{array}{#1}}
+    \def\endmatrix{\end{array}\right)}
+\def\smatrix{\left(\begin{smallmatrix}}
+    \def\endsmatrix{\end{smallmatrix}\right)}
+
+\def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{%
+    \expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup}
+    \expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
+    \expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
+    \expandafter\def\csname #1@ende\endcsname##1{#8}
+}
+\multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}}
+\multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}}
+\multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{}
+\multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}}
+
+\def\cases[#1]#2{\left\{\begin{array}[#1]{#2}}
+    \def\endcases{\end{array}\right.}
+
+\def\BeweisRichtung[#1]{\@ifnextchar\bgroup{\@BeweisRichtung@c[#1]}{\@BeweisRichtung@bes[#1]}}
+    \def\@BeweisRichtung@bes[#1]{{\bfseries(#1).~}}
+    \def\@BeweisRichtung@c[#1]#2#3{{\bfseries(#2#1#3).~}}
+\def\erzeugeBeweisRichtungBefehle#1#2{
+    \expandafter\gdef\csname #1text\endcsname##1##2{\BeweisRichtung[#2]{##1}{##2}}
+    \expandafter\gdef\csname #1\endcsname{%
+        \@ifnextchar\bgroup{\csname #1@\endcsname}{\csname #1text\endcsname{}{}}%
+    }
+    \expandafter\gdef\csname #1@\endcsname##1##2{%
+        \csname #1text\endcsname{\punktcref{##1}}{\punktcref{##2}}%
+    }
+}
+\erzeugeBeweisRichtungBefehle{hinRichtung}{$\Longrightarrow$}
+\erzeugeBeweisRichtungBefehle{herRichtung}{$\Longleftarrow$}
+\erzeugeBeweisRichtungBefehle{hinherRichtung}{$\Longleftrightarrow$}
+
+\def\cal#1{\mathcal{#1}}
+\def\brkt#1{\langle{}#1{}\rangle}
+\def\mathfrak#1{\mbox{\usefont{U}{euf}{m}{n}#1}}
+\def\kurs#1{\textit{#1}}
+\def\rectangleblack{\text{\RectangleBold}}
+\def\rectanglewhite{\text{\Rectangle}}
+\def\squareblack{\blacksquare}
+\def\squarewhite{\Box}
+
+%% ********************************************************************************
+%% FILE: src/setup-environments.tex
+%% ********************************************************************************
+
+%% **********************************************************************
+%% CLEVEREF: ************************************************************
+
+    \def\crefname@full#1#2#3{\crefname{#1}{#2}{#3}\Crefname{#1}{#2}{#3}}
+    \crefname@full{chapter}{Kapitel}{Kapitel}
+    \crefname@full{section}{Abschnitt}{Abschnitte}
+    \crefname@full{figure}{Fig.}{Fig.}
+    \crefname@full{subfigure}{Fig.}{Fig.}
+
+    \crefname@full{proof}{Beweis}{Beweise}
+    \crefname@full{thm}{Theorem}{Theoreme}
+    \crefname@full{satz}{Satz}{Sätze}
+    \crefname@full{claim}{Behauptung}{Behauptungen}
+    \crefname@full{lemm}{Lemma}{Lemmata}
+    \crefname@full{cor}{Korollar}{Korollarien}
+    \crefname@full{folg}{Folgerung}{Folgerungen}
+    \crefname@full{prop}{Proposition}{Propositionen}
+    \crefname@full{defn}{Definition}{Definitionen}
+    \crefname@full{conv}{Konvention}{Konventionen}
+    \crefname@full{fact}{Fakt}{Fakten}
+    \crefname@full{rem}{Bemerkung}{Bemerkungen}
+    \crefname@full{qstn}{Frage}{Fragen}
+    \crefname@full{e.g.}{Beipsiel}{Beipsiele}
+
+%% ****************************************************************
+%% THEOREME:
+%% ****************************************************************
+
+    \def\qedEIGEN#1{\@ifnextchar[{\qedEIGEN@c{#1}}{\qedEIGEN@bes{#1}}}%]
+    \def\qedEIGEN@bes#1{%
+        \bgroup%
+        \parfillskip=0pt%            % so \par doesnt push \square to left
+        \widowpenalty=10000%         % so we dont break the page before \square
+        \displaywidowpenalty=10000%  % ditto
+        \finalhyphendemerits=0%      % TeXbook exercise 14.32
+        \leavevmode%                 % \nobreak means lines not pages
+        \unskip%                     % remove previous space or glue
+        \nobreak%                    % don’t break lines
+        \hfil%                       % ragged right if we spill over
+        \penalty50%                  % discouragement to do so
+        \hskip.2em%                  % ensure some space
+        \null%                       % anchor following \hfill
+        \hfill%                      % push \square to right
+        #1%                          % the end-of-proof mark
+        \par%
+        \egroup%
+    }
+    \def\qedEIGEN@c#1[#2]{%
+        \bgroup%
+        \parfillskip=0pt%            % so \par doesnt push \square to left
+        \widowpenalty=10000%         % so we dont break the page before \square
+        \displaywidowpenalty=10000%  % ditto
+        \finalhyphendemerits=0%      % TeXbook exercise 14.32
+        \leavevmode%                 % \nobreak means lines not pages
+        \unskip%                     % remove previous space or glue
+        \nobreak%                    % don’t break lines
+        \hfil%                       % ragged right if we spill over
+        \penalty50%                  % discouragement to do so
+        \hskip.2em%                  % ensure some space
+        \null%                       % anchor following \hfill
+        \hfill%                      % push \square to right
+        {#1~{\smaller\bfseries\upshape (#2)}}%
+        \par%
+        \egroup%
+    }
+    \def\qedVARIANT#1#2{
+        \expandafter\def\csname ennde#1Sign\endcsname{#2}
+        \expandafter\def\csname ennde#1\endcsname{\@ifnextchar[{\qedEIGEN@c{#2}}{\qedEIGEN@bes{#2}}} %]
+    }
+    \qedVARIANT{OfProof}{$\squareblack$}
+    \qedVARIANT{OfWork}{\rectangleblack}
+    \qedVARIANT{OfSomething}{$\dashv$}
+    \qedVARIANT{OnNeutral}{$\lozenge$} % \lozenge \bigcirc \blacklozenge
+    \def\qedsymbol{\enndeOfProofSign}
+    \def\proofSymbol{\enndeOfProofSign}
+
+    \def\ra@pretheoremwork{
+        \setlength{\theorempreskipamount}{\ownspaceabovethm}
+    }
+    \def\rathmtransfer#1#2{
+        \expandafter\def\csname #2\endcsname{\csname #1\endcsname}
+        \expandafter\def\csname end#2\endcsname{\csname end#1\endcsname}
+    }
+
+    \def\ranewthm#1#2#3[#4]{
+        %% FOR \BEGIN{THM}
+        \theoremstyle{\current@theoremstyle}
+        \theoremseparator{\current@theoremseparator}
+        \theoremprework{\ra@pretheoremwork}
+        \@ifundefined{#1@basic}{\newtheorem{#1@basic}[#4]{#2}}{\renewtheorem{#1@basic}[#4]{#2}}
+        %% FOR \BEGIN{THM}[...]
+        \theoremstyle{\current@theoremstyle}
+        \theoremseparator{\thmForceSepPt}
+        \theoremprework{\ra@pretheoremwork}
+        \@ifundefined{#1@withName}{\newtheorem{#1@withName}[#4]{#2}}{\renewtheorem{#1@withName}[#4]{#2}}
+        %% FOR \BEGIN{THM*}
+        \theoremstyle{nonumberplain}
+        \theoremseparator{\thmForceSepPt}
+        \theoremprework{\ra@pretheoremwork}
+        \@ifundefined{#1@star@basic}{\newtheorem{#1@star@basic}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@basic}[Xdisplaynone]{#2}}
+        %% FOR \BEGIN{THM*}[...]
+        \theoremstyle{nonumberplain}
+        \theoremseparator{\thmForceSepPt}
+        \theoremprework{\ra@pretheoremwork}
+        \@ifundefined{#1@star@withName}{\newtheorem{#1@star@withName}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@withName}[Xdisplaynone]{#2}}
+        %% GENERATE ENVIRONMENTS:
+        \umbauenenv{#1}{#3}[#4]
+        \umbauenenv{#1@star}{#3}[Xdisplaynone]
+        %% TRANSFER *-DEFINITION
+        \rathmtransfer{#1@star}{#1*}
+    }
+
+    \def\umbauenenv#1#2[#3]{%
+        %% \BEGIN{THM}...
+        \expandafter\def\csname #1\endcsname{\relax%
+            \@ifnextchar[{\csname #1@\endcsname}{\csname #1@\endcsname[*]}%
+        }
+        %% \BEGIN{THM}[ANFANG]...
+        \expandafter\def\csname #1@\endcsname[##1]{\relax%
+            \@ifnextchar[{\csname #1@@\endcsname[##1]}{\csname #1@@\endcsname[##1][*]}%
+        }
+        %% \BEGIN{THM}[ANFANG][SCHLUSS]
+        \expandafter\def\csname #1@@\endcsname[##1][##2]{%
+            \ifx*##1%
+                \def\enndeOfBlock{\csname end#1@basic\endcsname}
+                \csname #1@basic\endcsname%
+            \else%
+                \def\enndeOfBlock{\csname end#1@withName\endcsname}
+                \csname #1@withName\endcsname[##1]%
+            \fi%
+            \def\makelabel####1{%
+                \gdef\beweislabel{####1}%
+                \label{\beweislabel}%
+            }%
+            \ifx*##2%
+                \def\enndeSymbol{\qedEIGEN{#2}}
+            \else%
+                \def\enndeSymbol{\qedEIGEN{#2}[##2]}
+            \fi
+        }
+        %% \END{THM}
+        \expandafter\gdef\csname end#1\endcsname{\enndeSymbol\enndeOfBlock}
+    }
+
+    %% NEWTHEOREM EINSTELLUNGSOPTIONEN:
+    %% F\"UR \theoremstyle
+    %% plain            Emulates original LATEX defin, except uses param \theorem...skipamount.
+    %% break            Header followed by line break.
+    %% change            Header, Number and Text are interchanged, without a line break.
+    %% changebreak        =change, but with a line break after Header.
+    %% margin            Number in left margin, without a line break.
+    %% marginbreak        =margin, but with a line break after the header.
+    %% nonumberplain    =plain, without number.
+    %% nonumberbreak    =break, without number.
+    %% empty            No number, no name. Only the optional argument is typeset.
+    %%    \theoremclass \theoremnumbering
+    %%    \theorempreskip \theorempostkip \theoremindent
+    %%    \theoremprework \theorempostwork
+
+        \def\current@theoremstyle{plain}
+        \def\current@theoremseparator{\thmnumberingseppt}
+        \theoremstyle{\current@theoremstyle}
+        \theoremseparator{\current@theoremseparator}
+        \theoremsymbol{}
+
+    \newtheorem{X}{X}[chapter] % for most theorems
+    \newtheorem{Xe}{Xe}[chapter] % for equations
+    \newtheorem*{Xdisplaynone}{Xdisplaynone}[chapter] % a dummy counter, that will never be displayed.
+    \newtheorem{Xsp}{Xsp}[chapter] % for special theorems
+    \generatenestedthmnumbering{arabic}{chapter}{X}
+    \generatenestedthmnumbering{arabic}{chapter}{Xe}
+    \generatenestedthmnumbering{Roman}{chapter}{Xsp}
+    \let\theXsp\theshortXsp
+
+        \theoremheaderfont{\upshape\bfseries}
+        \theorembodyfont{\slshape}
+
+    \ranewthm{thm}{Theorem}{\enndeOnNeutralSign}[X]
+    \ranewthm{satz}{Satz}{\enndeOnNeutralSign}[X]
+    \ranewthm{claim}{Behauptung}{\enndeOnNeutralSign}[X]
+    \ranewthm{lemm}{Lemma}{\enndeOnNeutralSign}[X]
+    \ranewthm{cor}{Korollar}{\enndeOnNeutralSign}[X]
+    \ranewthm{folg}{Folgerung}{\enndeOnNeutralSign}[X]
+    \ranewthm{prop}{Proposition}{\enndeOnNeutralSign}[X]
+
+        \theorembodyfont{\upshape}
+
+    \ranewthm{defn}{Definition}{\enndeOnNeutralSign}[X]
+    \ranewthm{conv}{Konvention}{\enndeOnNeutralSign}[X]
+    \ranewthm{obs}{Beobachtung}{\enndeOnNeutralSign}[X]
+    \ranewthm{e.g.}{Beipsiel}{\enndeOnNeutralSign}[X]
+    \ranewthm{fact}{Fakt}{\enndeOnNeutralSign}[X]
+    \ranewthm{rem}{Bemerkung}{\enndeOnNeutralSign}[X]
+    \ranewthm{qstn}{Frage}{\enndeOnNeutralSign}[X]
+    \ranewthm{exer}{Aufgabe}{\enndeOnNeutralSign}[X]
+    \ranewthm{soln}{Lösung}{\enndeOnNeutralSign}[X]
+
+        \theoremheaderfont{\itshape\bfseries}
+        \theorembodyfont{\upshape}
+
+    \ranewthm{proof@tmp}{Beweis}{\enndeOfProofSign}[Xdisplaynone]
+    \rathmtransfer{proof@tmp*}{proof}
+
+    \def\behauptungbeleg@claim{%
+        \iflanguage{british}{Claim}{%
+        \iflanguage{english}{Claim}{%
+        \iflanguage{ngerman}{Behauptung}{%
+        \iflanguage{russian}{Утверждение}{%
+        Claim%
+        }}}}%
+    }
+    \def\behauptungbeleg@pf@kurz{%
+        \iflanguage{british}{Pf}{%
+        \iflanguage{english}{Pf}{%
+        \iflanguage{ngerman}{Bew}{%
+        \iflanguage{russian}{Доказательство}{%
+        Pf%
+        }}}}%
+    }
+    \def\behauptungbeleg{\@ifnextchar\bgroup{\behauptungbeleg@c}{\behauptungbeleg@bes}}
+            \def\behauptungbeleg@c#1{\item[{\bfseries \behauptungbeleg@claim\erlaubeplatz #1.}]}
+            \def\behauptungbeleg@bes{\item[{\bfseries \behauptungbeleg@claim.}]}
+        \def\belegbehauptung{\item[{\bfseries\itshape\behauptungbeleg@pf@kurz.}]}
+
+%% ****************************************************************
+%% ALTE UMGEBUNGEN:
+%% ****************************************************************
+
+    \newcolumntype{\RECHTS}[1]{>{\raggedleft}p{#1}}
+    \newcolumntype{\LINKS}[1]{>{\raggedright}p{#1}}
+    \newcolumntype{m}{>{$}l<{$}}
+    \newcolumntype{C}{>{$}c<{$}}
+    \newcolumntype{L}{>{$}l<{$}}
+    \newcolumntype{R}{>{$}r<{$}}
+    \newcolumntype{0}{@{\hspace{0pt}}}
+    \newcolumntype{\LINKSRAND}{@{\hspace{\@totalleftmargin}}}
+    \newcolumntype{h}{@{\extracolsep{\fill}}}
+    \newcolumntype{i}{>{\itshape}}
+    \newcolumntype{t}{@{\hspace{\tabcolsep}}}
+    \newcolumntype{q}{@{\hspace{1em}}}
+    \newcolumntype{n}{@{\hspace{-\tabcolsep}}}
+    \newcolumntype{M}[2]{%
+        >{\begin{minipage}{#2}\begin{math}}%
+        {#1}%
+        <{\end{math}\end{minipage}}%
+    }
+    \newcolumntype{T}[2]{%
+        >{\begin{minipage}{#2}}%
+        {#1}%
+        <{\end{minipage}}%
+    }
+    \setlength{\LTpre}{\baselineskip}
+    \setlength{\LTpost}{0pt}
+    \def\center{\centering}
+        \def\endcenter{}
+
+    \def\punkteumgebung@genbefehl#1#2#3{
+        \punkteumgebung@genbefehl@{#1}{#2}{#3}{}{}
+        \punkteumgebung@genbefehl@{multi#1}{#2}{#3}{
+            \setlength{\columnsep}{10pt}%
+            \setlength{\columnseprule}{0pt}%
+            \begin{multicols}{\thecolumnanzahl}%
+        }{\end{multicols}\nvraum{1}}
+    }
+    \def\punkteumgebung@genbefehl@#1#2#3#4#5{
+        \expandafter\gdef\csname #1\endcsname{
+            \@ifnextchar\bgroup{\csname #1@c\endcsname}{\csname #1@bes\endcsname}
+        }%]
+            \expandafter\def\csname #1@c\endcsname##1{
+                \@ifnextchar[{\csname #1@c@\endcsname{##1}}{\csname #1@c@\endcsname{##1}[\z@]}
+            }%]
+            \expandafter\def\csname #1@c@\endcsname##1[##2]{
+                \@ifnextchar[{\csname #1@c@@\endcsname{##1}[##2]}{\csname #1@c@@\endcsname{##1}[##2][\z@]}
+            }%]
+            \expandafter\def\csname #1@c@@\endcsname##1[##2][##3]{
+                \let\alterlinkerRand\gesamtlinkerRand
+                \let\alterrechterRand\gesamtrechterRand
+                \addtolength{\gesamtlinkerRand}{##2}
+                \addtolength{\gesamtrechterRand}{##3}
+                \advance\linewidth -##2%
+                \advance\linewidth -##3%
+                \advance\@totalleftmargin ##2%
+                \parshape\@ne \@totalleftmargin\linewidth%
+                #4
+                \begin{#2}[\upshape ##1]%
+                    \setlength{\parskip}{0.5\baselineskip}\relax%
+                    \setlength{\topsep}{\z@}\relax%
+                    \setlength{\partopsep}{\z@}\relax%
+                    \setlength{\parsep}{\parskip}\relax%
+                    \setlength{\itemsep}{#3}\relax%
+                    \setlength{\listparindent}{\z@}\relax%
+                    \setlength{\itemindent}{\z@}\relax%
+            }
+            \expandafter\def\csname #1@bes\endcsname{
+                \@ifnextchar[{\csname #1@bes@\endcsname}{\csname #1@bes@\endcsname[\z@]}
+            }%]
+            \expandafter\def\csname #1@bes@\endcsname[##1]{
+                \@ifnextchar[{\csname #1@bes@@\endcsname[##1]}{\csname #1@bes@@\endcsname[##1][\z@]}
+            }%]
+            \expandafter\def\csname #1@bes@@\endcsname[##1][##2]{
+                \let\alterlinkerRand\gesamtlinkerRand
+                \let\alterrechterRand\gesamtrechterRand
+                \addtolength{\gesamtlinkerRand}{##1}
+                \addtolength{\gesamtrechterRand}{##2}
+                \advance\linewidth -##1%
+                \advance\linewidth -##2%
+                \advance\@totalleftmargin ##1%
+                \parshape\@ne \@totalleftmargin\linewidth%
+                #4
+                \begin{#2}%
+                    \setlength{\parskip}{0.5\baselineskip}\relax%
+                    \setlength{\topsep}{\z@}\relax%
+                    \setlength{\partopsep}{\z@}\relax%
+                    \setlength{\parsep}{\parskip}\relax%
+                    \setlength{\itemsep}{#3}\relax%
+                    \setlength{\listparindent}{\z@}\relax%
+                    \setlength{\itemindent}{\z@}\relax%
+            }
+        \expandafter\gdef\csname end#1\endcsname{%
+            \end{#2}#5
+            \setlength{\gesamtlinkerRand}{\alterlinkerRand}
+            \setlength{\gesamtlinkerRand}{\alterrechterRand}
+        }
+    }
+
+    \def\ritempunkt{{\Large\textbullet}} % \textbullet, $\sqbullet$, $\blacktriangleright$
+    \setdefaultitem{\ritempunkt}{\ritempunkt}{\ritempunkt}{\ritempunkt}
+    \punkteumgebung@genbefehl{itemise}{compactitem}{\parskip}{}{}
+    \punkteumgebung@genbefehl{kompaktitem}{compactitem}{\z@}{}{}
+    \punkteumgebung@genbefehl{enumerate}{compactenum}{\parskip}{}{}
+    \punkteumgebung@genbefehl{kompaktenum}{compactenum}{\z@}{}{}
+
+    \let\ALTthebibliography\thebibliography
+    \renewenvironment{thebibliography}[1]{%
+        \begin{ALTthebibliography}{#1}
+        \addcontentsline{toc}{part}{\bibname}
+    }{%
+        \end{ALTthebibliography}
+    }
+
+%% ****************************************************************
+%% NEUE UMGEBUNGEN:
+%% ****************************************************************
+
+    \def\matrix#1{\left(\begin{array}[mc]{#1}}
+        \def\endmatrix{\end{array}\right)}
+    \def\smatrix{\left(\begin{smallmatrix}}
+        \def\endsmatrix{\end{smallmatrix}\right)}
+    \def\vector{\begin{matrix}{c}}
+        \def\endvector{\end{matrix}}
+    \def\svector{\begin{smatrix}}
+        \def\endsvector{\end{smatrix}}
+
+    \def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{%
+        \expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup}
+        \expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
+        \expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
+        \expandafter\def\csname #1@ende\endcsname##1{#8}
+    }
+    \multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}}
+    \multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}}
+    \multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{}
+    \multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}}
+
+    \def\underbracenodisplay#1{%
+        \mathop{\vtop{\m@th\ialign{##\crcr
+        $\hfil\displaystyle{#1}\hfil$\crcr
+        \noalign{\kern3\p@\nointerlineskip}%
+        \upbracefill\crcr\noalign{\kern3\p@}}}}\limits%
+    }
+
+    \def\mathe[#1]#2{%
+        \ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\begin{escapeeinzug}}
+        \noindent%
+        \let\eqtagset\boolfalsch
+        \let\eqtaglabel\boolleer
+        \let\eqtagsymb\boolleer
+        \let\alteqtag\eqtag
+        \def\eqtag{\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}}%
+        \def\eqtag@loc@[##1]{\@ifnextchar\bgroup{\eqtag@loc@@[##1]}{\eqtag@loc@@[##1]{}}}%
+        \def\eqtag@loc@@[##1]##2{%
+            \gdef\eqtagset{\boolwahr}
+            \gdef\eqtaglabel{##1}
+            \gdef\eqtagsymb{##2}
+        }%
+        \def\verticalalign{}%
+            \IfBeginWith{#1}{t}{\def\verticalalign{t}}{}%
+            \IfBeginWith{#1}{m}{\def\verticalalign{c}}{}%
+            \IfBeginWith{#1}{b}{\def\verticalalign{b}}{}%
+        \def\horizontalalign{\null\hfill\null}%
+            \IfEndWith{#1}{l}{}{\null\hfill\null}%
+            \IfEndWith{#1}{r}{\def\horizontalalign{}}{}%
+        \begin{math}
+        \begin{array}[\verticalalign]{0#2}%
+    }
+        \def\endmathe{%
+            \end{array}
+            \end{math}\horizontalalign%
+            \let\eqtag\alteqtag
+            \ifthenelse{\equal{\eqtagset}{\boolwahr}}{\eqtag[\eqtaglabel]{\eqtagsymb}}{}
+            \ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\end{escapeeinzug}}%
+        }
+
+    \def\longmathe[#1]#2{\relax
+        \let\altarraystretch\arraystretch
+        \renewcommand\arraystretch{1.2}\relax
+        \begin{longtable}[#1]{\LINKSRAND #2}
+    }
+        \def\endlongmathe{
+            \end{longtable}
+            \renewcommand\arraystretch{\altarraystretch}
+        }
+
+    \def\einzug{\@ifnextchar[{\indents@}{\indents@[\z@]}}%]
+        \def\indents@[#1]{\@ifnextchar[{\indents@@[#1]}{\indents@@[#1][\z@]}}%]
+        \def\indents@@[#1][#2]{%
+            \begin{list}{}{\relax
+                \setlength{\topsep}{\z@}\relax
+                \setlength{\partopsep}{\z@}\relax
+                \setlength{\parsep}{\parskip}\relax
+                \setlength{\listparindent}{\z@}\relax
+                \setlength{\itemindent}{\z@}\relax
+                \setlength{\leftmargin}{#1}\relax
+                \setlength{\rightmargin}{#2}\relax
+                \let\alterlinkerRand\gesamtlinkerRand
+                \let\alterrechterRand\gesamtrechterRand
+                \addtolength{\gesamtlinkerRand}{#1}
+                \addtolength{\gesamtrechterRand}{#2}
+            }\relax
+                \item[]\relax
+        }
+            \def\endeinzug{%
+                \setlength{\gesamtlinkerRand}{\alterlinkerRand}
+                \setlength{\gesamtlinkerRand}{\alterrechterRand}
+                \end{list}%
+            }
+
+    \def\escapeeinzug{\begin{einzug}[-\gesamtlinkerRand][-\gesamtrechterRand]}
+        \def\endescapeeinzug{\end{einzug}}
+
+    \def\programmiercode{
+        \modulolinenumbers[1]
+        \begin{einzug}[\rtab][\rtab]%
+        \begin{linenumbers}%
+            \fontfamily{cmtt}\fontseries{m}\fontshape{u}\selectfont%
+            \setlength{\parskip}{1\baselineskip}%
+            \setlength{\parindent}{0pt}%
+    }
+        \def\endprogrammiercode{
+            \end{linenumbers}
+            \end{einzug}
+        }
+
+    \def\schattiertebox@genbefehl#1#2#3{
+        \expandafter\gdef\csname #1\endcsname{%
+            \@ifnextchar[{\csname #1@args\endcsname}{\csname #1@args\endcsname[#3]}%]%
+        }
+            \expandafter\def\csname #1@args\endcsname[##1]{%
+                \@ifnextchar[{\csname #1@args@l\endcsname[##1]}{\csname #1@args@n\endcsname[##1]}%]%
+            }
+            \expandafter\def\csname #1@args@l\endcsname[##1][##2]{%
+                \@ifnextchar[{\csname #1@args@l@r\endcsname[##1][##2]}{\csname #1@args@l@n\endcsname[##1][##2]}%]%
+            }
+            \expandafter\def\csname #1@args@n\endcsname[##1]{%
+                \let\boolinmdframed\boolwahr
+                \begin{mdframed}[#2leftmargin=0,rightmargin=0,outermargin=0,innermargin=0,##1]
+            }
+            \expandafter\def\csname #1@args@l@n\endcsname[##1][##2]{%
+                \let\boolinmdframed\boolwahr
+                \begin{mdframed}[#2leftmargin=##2/2,rightmargin=##2/2,outermargin=##2/2,innermargin=##2/2,##1]
+            }
+            \expandafter\def\csname #1@args@l@r\endcsname[##1][##2][##3]{%
+                \let\boolinmdframed\boolwahr
+                \begin{mdframed}[#2leftmargin=##2,rightmargin=##3,outermargin=##2,innermargin=##3,##1]
+            }
+        \expandafter\gdef\csname end#1\endcsname{%
+            \end{mdframed}
+            \let\boolinmdframed\boolfalsch
+        }
+    }
+        \schattiertebox@genbefehl{schattiertebox}{
+            splittopskip=0,%
+            splitbottomskip=0,%
+            frametitleaboveskip=0,%
+            frametitlebelowskip=0,%
+            skipabove=1\baselineskip,%
+            skipbelow=1\baselineskip,%
+            linewidth=2pt,%
+            linecolor=black,%
+            roundcorner=4pt,%
+        }{
+            backgroundcolor=leer,%
+            nobreak=true,%
+        }
+
+        \schattiertebox@genbefehl{schattierteboxdunn}{
+            splittopskip=0,%
+            splitbottomskip=0,%
+            frametitleaboveskip=0,%
+            frametitlebelowskip=0,%
+            skipabove=1\baselineskip,%
+            skipbelow=1\baselineskip,%
+            linewidth=1pt,%
+            linecolor=black,%
+            roundcorner=2pt,%
+        }{
+            backgroundcolor=leer,%
+            nobreak=true,%
+        }
+
+        \def\algorithm{\schattiertebox[backgroundcolor=hellgrau,nobreak=false]}
+            \def\endalgorithm{\endschattiertebox}
+
+    \def\tikzsetzenode#1{%
+        \tikz[remember picture,baseline,overlay]{\node #1;}%
+    }
+    \def\tikzsetzepfeil#1{%
+        \begin{tikzpicture}[remember picture,overlay,>=latex]%
+            \draw #1;%
+        \end{tikzpicture}%
+    }
+    \def\tikzsetzeoverlay#1{%
+        \begin{tikzpicture}[remember picture,overlay,>=latex]%
+            #1%
+        \end{tikzpicture}%
+    }
+    \def\tikzsetzekreise[#1]#2#3{%
+        \tikzsetzepfeil{%
+        [rounded corners,#1]%
+            ([shift={(-\tabcolsep,0.75\baselineskip)}]#2)%
+            rectangle%
+            ([shift={(\tabcolsep,-0.5\baselineskip)}]#3)
+        }%
+    }
+
+    \tikzset{
+        >=stealth,
+        auto,
+        thick,
+        main node/.style={
+            circle,draw,font=\sffamily\Large\bfseries,minimum size=0pt
+        },
+    }
+
+%% ********************************************************************************
+%% FILE: src/setup-layout.tex
+%% ********************************************************************************
+
+\pagestyle{fancyplain}
+
+\@ifundefined{setcitestyle}{%
+    %% do nothing
+}{%
+    \setcitestyle{numeric-comp,open={[},close={]}}
+}
+\def\crefpairconjunction{ und }
+\def\crefmiddleconjunction{, }
+\def\creflastconjunction{, und }
+
+\raggedbottom %% <- pushes footers up
+\sloppy
+\def\headrulewidth{0pt}
+\def\footrulewidth{0pt}
+\setlength{\columnsep}{20pt}
+\setlength{\columnseprule}{1pt}
+\setlength{\headheight}{11pt}
+\setlength{\partopsep}{0pt}
+\setlength{\topsep}{\baselineskip}
+\setlength{\topskip}{0.5\baselineskip}
+\setlength{\footskip}{-1\baselineskip}
+\setlength{\maxdepth}{0pt}
+\renewcommand{\baselinestretch}{1}
+\renewcommand{\arraystretch}{1}
+\setcounter{LTchunksize}{\infty}
+\setlength{\abovedisplayskip}{0pt}
+\setlength{\parskip}{1\baselineskip}
+\def\firstparagraph{\noindent}
+\def\continueparagraph{\noindent}
+
+\hypersetup{
+    hidelinks=true,
+}
+
+    \@addtoreset{chapter}{part} %% nötig für Hyperref.
+
+    \def\partfont{\documentfont\fontseries{bx}\Huge\selectfont}
+    \def\chapterfont{\documentfont\fontseries{bx}\huge\selectfont}
+    \def\sectionfont{\documentfont\fontseries{bx}\Large\selectfont}
+    \def\subsectionfont{\documentfont\fontseries{bx}\large\selectfont}
+
+    \def\thepart{\Roman{part}}
+    \generatenestedsecnumbering{arabic}{part}{chapter}
+    \generatenestedsecnumbering{arabic}{chapter}{section}
+    \generatenestedsecnumbering{arabic}{section}{subsection}
+    \generatenestedsecnumbering{arabic}{subsection}{subsubsection}
+    \def\theunitnamepart{\thepart}
+    \def\theunitnamechapter{\theshortchapter}
+    \def\theunitnamesection{\thelongsection}
+    \def\theunitnamesubsection{\thelongsubsection}
+    \def\theunitnamesubsubsection{\thelongsubsubsection}
+
+    \def\partname{Teil\erlaubeplatz}
+    \def\chaptername{Kapitel\erlaubeplatz}
+    \def\sectionname{\S\erlaubeplatz}
+    \def\subsectionname{}
+    \def\subsubsectionname{}
+
+    \let\appendix@orig\appendix
+    \def\appendix{%
+        \appendix@orig%
+        \let\boolinappendix\boolwahr
+        \addcontentsline{toc}{part}{\appendixname}%
+        \addtocontents{toc}{\protect\setcounter{tocdepth}{0}}
+        \def\sectionname{Appendix}%
+        \def\theunitnamesection{\Alph{section}}%
+    }
+    \def\notappendix{%
+        \let\boolinappendix\boolfalse
+        \addtocontents{toc}{\protect\setcounter{tocdepth}{1 }}
+        \def\sectionname{}%
+        \def\theunitnamesection{\arabic{section}}%
+    }
+
+%% \titlespacing{<sectionclassname>}
+%%	{linker einzug}{platz oberhalb}{platz unterhalb}[rechter einzug]
+
+    \titlespacing{\section}{0pt}{\baselineskip}{\baselineskip}
+    \titlespacing{\subsection}{0pt}{\baselineskip}{\baselineskip}
+    \titlespacing{\subsubsection}{0pt}{\baselineskip}{\baselineskip}
+    \titlespacing{\paragraph}{0pt}{0pt}{1em}
+
+    \titleformat{\part}[display]
+        {\normalfont\headingfont\bfseries\Huge\centering}
+        {%
+            \ifthenelse{\equal{\partname}{}}{%
+                \theunitnamepart%
+            }{%
+                \MakeUppercase{\partname}~\theunitnamepart%
+            }%
+        }{0pt}{%
+        }[\thispagestyle{empty}]
+    \titleformat{\chapter}[frame]
+        {\normalfont\headingfont\bfseries\Large}
+        {%
+            \bedingtesspaceexpand{chaptername}{~}{\theunitnamechapter}%
+        }{0.5em}{%
+        }[\thispagestyle{empty}]%\titlerule%[2pt]%
+    \titleformat{\section}[hang]
+        {\normalfont\headingfont\bfseries\flushleft\large}
+        {%
+            \bedingtesspaceexpand{sectionname}{~}{\theunitnamesection}%
+        }{0.5em}
+        {%
+        }
+        [%
+            \nvraum{0.25}%
+        ]
+    \titleformat{\subsection}[hang]
+        {\normalfont\headingfont\bfseries\flushleft\large}
+        {%
+            \bedingtesspaceexpand{subsectionname}{~}{\theunitnamesubsection}%
+        }{0.5em}
+        {%
+        }
+        [%
+            \nvraum{0.25}%
+        ]
+    \titleformat{\subsubsection}[hang]
+        {\normalfont\headingfont\bfseries\flushleft\large}
+        {%
+            \bedingtesspaceexpand{subsubsectionname}{~}{\theunitnamesubsubsection}%
+        }{0.5em}
+        {%
+        }
+        [%
+            \nvraum{0.25}%
+        ]
+
+\def\rafootnotectr{20}
+\def\incrftnotectr#1{%
+    \addtocounter{#1}{1}%
+    \ifnum\value{#1}>\rafootnotectr\relax
+        \setcounter{#1}{0}%
+    \fi%
+}
+\def\footnoteref[#1]{\protected@xdef\@thefnmark{\ref{#1}}\@footnotemark}
+\let\altfootnotetext\footnotetext
+    \def\footnotetext[#1]#2{\incrftnotectr{footnote}\altfootnotetext[\value{footnote}]{\label{#1}#2}}
+\let\altfootnotemark\footnotemark
+    %% Undesirable solution, as the text is not hyperlinked.
+    \def\footnotemark[#1]{\text{\textsuperscript{\getrefnumber{#1}}}}
+
+\DefineFNsymbols*{custom}{abcdefghijklmnopqrstuvwxyz}
+\setfnsymbol{custom}
+\def\footnotelayout{\documentfont\scriptsize}
+\def\thefootnote{\fnsymbol{footnote}}
+
+\def\kopfzeileleer{
+    \lhead[]{}
+    \chead[]{}
+    \rhead[]{}
+    \lfoot[]{}
+    \cfoot[]{}
+    \rfoot[]{}
+}
+\def\kopfzeiledefault{
+    \lhead[]{}
+    \lhead[]{}
+    \chead[]{}
+    \rhead[]{}
+    \lfoot[]{}
+    \cfoot{\footnotesize\thepage}
+    \rfoot[]{}
+}
+
+\DeclareRobustCommand\crfamily{\fontfamily{pcr}\selectfont}
+\def\headingfont{\fontfamily{cmss}\selectfont}
+\def\documentfancyfont{%
+    \gdef\headingfont{\crfamily}%
+    \fontfamily{ccr}\fontseries{m}\selectfont%
+}
+\def\documentfont{%
+    \gdef\headingfont{\fontfamily{cmss}\selectfont}%
+    \fontfamily{cmss}\fontseries{m}\selectfont%
+    \renewcommand{\sfdefault}{phv}%
+    \renewcommand{\ttdefault}{pcr}%
+    \renewcommand{\rmdefault}{cmr}% <— funktionieren nicht mit {ptm}
+    \renewcommand{\bfdefault}{bx}%
+    \renewcommand{\itdefault}{it}%
+    \renewcommand{\sldefault}{sl}%
+    \renewcommand{\scdefault}{sc}%
+    \renewcommand{\updefault}{n}%
+}
+
+\allowdisplaybreaks
+\let\altcleardoublepage\cleardoublepage
+\let\cleardoublepage\clearpage
+
+\def\startdocumentlayoutoptions{
+    \selectlanguage{ngerman}
+    \setlength{\parskip}{0.5\baselineskip}
+    \setlength{\parindent}{0pt}
+    \kopfzeiledefault
+    \documentfont
+    \normalsize
+}
+
+\def\highlightTerm#1{\emph{#1}}
+
+%% ********************************************************************************
+%% FILE: srclocal/setup-locallayout.tex
+%% ********************************************************************************
+
+\def\theunitnamesection{\theshortsection}
+
+%% ********************************************************************************
+%% FILE: srclocal/setup-localmacros.tex
+%% ********************************************************************************
+
+%% ****************************************************************
+%% MATHE:
+%% ****************************************************************
+
+\def\cal#1{\mathcal{#1}}
+\def\reell{\mathbb{R}}
+\def\kmplx{\mathbb{C}}
+\def\Torus{\mathbb{T}}
+\def\rtnl{\mathbb{Q}}
+\def\intgr{\mathbb{Z}}
+
+\def\ntrl{\mathbb{N}}
+\def\ntrlpos{\mathbb{N}}
+\def\ntrlzero{\mathbb{N}_{0}}
+\def\reellNonNeg{\reell_{+}}
+
+\def\imageinh{\imath}
+\def\ReTeil{\mathop{\mathfrak{R}\text{\upshape e}}}
+\def\ImTeil{\mathop{\mathfrak{I}\text{\upshape m}}}
+
+\def\leer{\emptyset}
+\def\restr#1{\vert_{#1}}
+\def\ohne{\mathbin{\setminus}}
+\def\Pot{\mathop{\mathcal{P}}}
+\def\einser{\mathbf{1}}
+\def\supp{\mathop{\mathrm{supp}}}
+
+\def\brkt#1{\langle{}#1{}\rangle}
+\def\lsim{\mathop{\sim}}
+\def\lneg{\mathop{\neg}}
+\def\land{\mathop{\wedge}}
+\def\lor{\mathop{\vee}}
+
+\def\eps{\varepsilon}
+\let\altphi\phi
+\let\altvarphi\varphi
+    \def\phi{\altvarphi}
+    \def\varphi{\altphi}
+
+\def\vectorspacespan{\mathop{\text{\upshape Lin}}}
+\def\dim{\mathop{\text{\upshape dim}}}
+\def\det{\mathop{\text{\upshape det}}}
+\def\rank{\mathop{\text{\upshape Rang}}}
+\def\onematrix{\text{\upshape\bfseries I}}
+\def\zeromatrix{\text{\upshape\bfseries 0}}
+\def\zerovector{\text{\upshape\bfseries 0}}
+
+\def\graph{\mathop{\text{\upshape Gph}}}
+\def\domain{\mathop{\text{\upshape dom}}}
+\def\range{\mathop{\text{\upshape Bild}}}
+\def\ker{\mathop{\text{\upshape Kern}}}
+\def\functionspace{\mathop{\text{\upshape Abb}}}
+\def\id{\text{\upshape id}}
+\def\modfn{\mathop{\text{\upshape mod}}}
+\def\divides{\mathbin{\mid}}
+\def\ndivides{\mathbin{\nmid}}
+\def\ggT{\mathop{\text{\upshape ggT}}}
+\def\choose#1#2{\begin{smatrix}#1\\#2\\\end{smatrix}}
+
+\def\punktschema{
+    \footnotesize
+    \hraum
+    \begin{tabular}[mc]{|p{0.1\textwidth}|p{0.7\textwidth}|}
+        \hline
+            \multicolumn{2}{|l|}{{\bfseries NOTENSCHEMA}}\\
+            Punkte &Beschreibung\\
+        \hline
+        \hline
+}
+    \def\endpunktschema{
+        \hline
+        \end{tabular}
+        \hraum
+    }
+
+\def\headingTeilaufgabe#1{
+    \uwave{{\bfseries\large Aufgabe #1}}
+}
+
+\makeatother
+
+\begin{document}
+    \startdocumentlayoutoptions
+
+    %% HAUPTTEXT:
+
+%% ********************************************************************************
+%% FILE: body/index.tex
+%% ********************************************************************************
+
+%% ********************************************************************************
+%% FILE: body/A5.tex
+%% ********************************************************************************
+
+%% AUFGABE 5
+\setcounter{chapter}{5}
+\let\altsectionname\sectionname
+\def\sectionname{Aufgabe}
+\setcounternach{section}{5}
+\section[]{(Klausur${}_{2}$, WiSe 2020/2021)}
+    \label{ueb:5}
+\let\sectionname\altsectionname
+
+Es sei $V$ ein endlich dimensionaler Vektorraum über einem Körper, $K$.
+Gegeben ist eine lineare Abbildung ${\phi:V\to V}$
+mit der Eigenschaft $\phi\circ\phi=\phi$.
+
+%% ********************************************************************************
+%% FILE: body/A5a.tex
+%% ********************************************************************************
+
+%% AUFGABE 5a
+\headingTeilaufgabe{5a}
+
+\begin{claim*}
+    $\ker(\phi)\cap\range(\phi)=\{\zerovector\}$.
+\end{claim*}
+
+Es gibt hierfür ein paar Ansätze.
+Zuerst der direkte Ansatz:
+
+\begin{enumerate}{\bfseries {Ansatz} I.}
+    \setcounternach{enumi}{1}
+    \item
+        \textbf{Zu zeigen:}
+
+        \begin{kompaktenum}{\bfseries (i)}
+            \item triviale Inklusion: $\{\zerovector\}\subseteq\ker(\phi)\cap\range(\phi)$;
+            \item nichttriviale Inklusion: $\ker(\phi)\cap\range(\phi)\subseteq\{\zerovector\}$.
+        \end{kompaktenum}
+
+        Zu (i):
+        Da $\ker(\phi)$ und $\range(\phi)$ Untervektorräume sind, liegt der Nullvektor in beiden und somit im Schnitt.
+
+        Zu (ii): Sei $v$ ein Vektor im Schnitt. Wir müssen zeigen, dass $v=\zerovector$.\\
+        Aus $v\in\range(\phi)$ folgt $v=\phi(w)$ für ein $w\in V$.\\
+        Da \uline{auch} $v\in\ker(\phi)$, erhalten wir
+
+            \begin{mathe}[mc]{rclql}
+                v &= &\phi(w)
+                    &\text{per Wahl von $w$}\\
+                &= &\phi(\phi(w))
+                    &\text{da per Annahme $\phi\circ\phi=\phi$}\\
+                &= &\phi(v)
+                    &\text{per Wahl von $w$}\\
+                &= &\zerovector
+                    &\text{da $v\in\ker(\phi)$.}\\
+            \end{mathe}
+
+        Also gelten (i) + (ii), woraus sich die behauptete Gleichung ergibt.
+        \enndeOfProof
+\end{enumerate}
+
+Für die folgenden Ansätze benutzen wir das Resultat aus (b).\\
+\textbf{ACHTUNG:} Dafür aber dürfen wir später (b) das Ergebnis aus (a) \uline{nicht} verwenden,
+denn sonst wäre dies ein \textbf{Zirkelschluss}.
+
+\begin{enumerate}{\bfseries {Ansatz} I.}
+    \setcounternach{enumi}{2}
+    \item
+        Laut (b) gilt ($\ast$)~$V=\ker(\phi)+\range(\phi)$.\\
+        Fixiere die Untervektorräume, $U_{1}:=\ker(\phi)$ und $U_{2}:=\range(\phi)$, von $V$.\\
+        Laut der \emph{Dimensionsformel für lineare Abbildungen} gilt:
+
+            \begin{mathe}[mc]{rcl}
+                \eqtag[eq:A5a:ansatz:2:1]
+                \dim(V) &= &\dim(\ker(\phi)) + \dim(\range(\phi)).\\
+            \end{mathe}
+
+        Aus ($\ast$) und der \emph{Dimensionsformel für Untervektorräume}
+        und da $V$ endlich dimensional ist,
+        erhalten wir außerdem:
+
+            \begin{mathe}[mc]{rcl}
+                \eqtag[eq:A5a:ansatz:2:2]
+                \dim(V) &= &\dim(\ker(\phi) + \range(\phi))\\
+                    &= &\dim(U_{1} + U_{2})\\
+                    &= &\dim(U_{1}) + \dim(U_{2}) - \dim(U_{1}\cap U_{2})\\
+                    &= &\dim(\ker(\phi)) + \dim(\range(\phi))
+                        - \dim(\ker(\phi)\cap\range(\phi)).\\
+            \end{mathe}
+
+        Da all diese Zahlen endlich sind, folgt aus
+            \eqcref{eq:A5a:ansatz:2:1}
+            und
+            \eqcref{eq:A5a:ansatz:2:2},
+        dass $\dim(\ker(\phi)\cap\range(\phi))=0$.\\
+        Da $\ker(\phi)\cap\range(\phi)$ ein Untervektorraum von $V$ ist
+        und $0$-dimensional ist,
+        gilt ${\ker(\phi)\cap\range(\phi)=\{\zerovector\}}$.
+        \enndeOfProof
+    \item
+        Laut (b) gilt $V=\ker(\phi)+\range(\phi)$.\\
+        Fixiere die Untervektorräume, $U_{1}:=\ker(\phi)$ und $U_{2}:=\range(\phi)$, von $V$.
+        Auf $U_{1}$ verhält sich $\phi$ wie $\zeromatrix$ (die Nullabbildung)
+        und auf $U_{2}$ verhält sich $\phi$ wie $\onematrix$ (die Identitätsabbildung),
+        weil $\phi\circ\phi=\phi$.
+
+        Für $x\in\ker(\phi)\cap\range(\phi)$ gilt $x\in U_{1}\cap U_{2}$.
+        Da sich $\phi$ wie die Null- bzw. Identitätsabbildung auf $U_{1}$ bzw. $U_{2}$ verhält,
+        gilt $x=\onematrix(x)=\phi(x)=\zeromatrix(x)=\zerovector$.
+        Also $\ker(\phi)\cap\range(\phi)\subseteq\{\zerovector\}$.
+        Da außerdem $\zerovector$ in allen Untervektorräumen liegt,
+        erhalten wir ${\ker(\phi)\cap\range(\phi)=\{\zerovector\}}$.
+        \enndeOfProof
+\end{enumerate}
+
+\begin{punktschema}
+    5   &Argument vollständig (=ausführlich) und logisch gültig.\\
+    \hline
+    4   &Nichttriviale Inklusion wurde vollständig bewiesen, aber die triviale Inklusion fehlte.\\
+    \hdashline
+    4   &Ansatz II/III gemacht aber fehlte zu erwähnen, dass $V$ endlich dimensional.\\
+    \hline
+    3   &Triviale Inklusion bewiesen und nichttriviale Inklusion lückenhaft (aber richtig) bewiesen.\\
+    \hdashline
+    3   &Argument baut auf falscher Annahme auf, aber Annahme „rechtfertigt“.\\
+    \hdashline
+    3   &Ansatz II/III gemacht, aber mehrere lücken.\\
+    \hline
+    2   &Nur triviale Inklusion bewiesen.\\
+    \hdashline
+    2   &Argument baut auf falscher Annahme auf und Annahme kaum rechtfertigt.\\
+    \hline
+    0   &sonst.\\
+\end{punktschema}
+
+{\footnotesize
+    \textbf{Bemerkungen.}
+    Da es sich hier um die Bewertung von Argumentationen handelt,
+    kann man in Wirklichkeit kein Schema festlegen.
+    Stattdessen musste über die Qualität Urteile getroffen werden.
+    In erster Linie kriegt man volle Punkte, wenn man
+    vollständig (idealerweise auch ausführlich) + gültig + überzeugend
+    argumentierte.
+    Ab dann wurden anhand unterschiedlicher Defizite empirische Graduierungen implementiert.
+    Wegen wichtiger Lücken wurden Punkte abgezogen.
+    Wenn man auf einer falschen Annahme aufbaute
+        (z.\,B. $\phi$ sei injektiv/bijektiv od. $\phi=\onematrix$ od. in Ansatz III eine falsche Dichotomie)
+    war entscheiden, ob man zumindest versuchte,
+    die falsche Annahme zu rechtfertigen.
+    Wenn man auf einer falschen Annahme das Argument aufbaute
+    und diese Annahme alles trivial machen ließ
+    und diese Annahme nicht genügend rechtfertigte,
+    dann resultierte dies in $0$ Punkten.
+    Wenn man Ansatz II/III implementierte aber im (b) mit dem Ergebnis aus (a) argumentierte,
+    war dies ein Kreisargument und somit wurden auch Punkte abgezogen.
+}
+
+\clearpage
+
+%% ********************************************************************************
+%% FILE: body/A5b.tex
+%% ********************************************************************************
+
+%% AUFGABE 5b
+\headingTeilaufgabe{5b}
+
+\begin{claim*}
+    Es gilt $V=\ker(\phi)+\range(\phi)$.
+\end{claim*}
+
+Es gab hier grundsätzlich zwei Ansätze:
+
+\begin{enumerate}{\bfseries {Ansatz} I.}
+    \item
+        \textbf{Zu zeigen:}
+
+        \begin{kompaktenum}{\bfseries (i)}
+            \item triviale Inklusion: ${\ker(\phi)+\range(\phi)\subseteq V}$;
+            \item nichttriviale Inklusion: ${V\subseteq\ker(\phi)+\range(\phi)}$.
+        \end{kompaktenum}
+
+        Zu (i):
+        Da $\ker(\phi),\range(\phi)\subseteq V$ und $V$ unter Addition abgeschlossen ist,
+        gilt ${\ker(\phi)+\range(\phi)\subseteq V}$.
+        (Alternative: die Summe aus Untervektorräumen ist i.\,A. ein Untervektorraum.)
+
+        Zu (ii):
+        Sei $v\in V$ beliebig.
+        Beachte zunächst, dass $v-\phi(v)\in\ker(\phi)$, da
+
+            \begin{mathe}[mc]{rcccl}
+                \phi(v-\phi(v)) &= &\phi(v)-\phi(\phi(v)) &\textoverset{($\ast$)}{=} &\phi(v) - \phi(v).\\
+            \end{mathe}
+
+        wobei ($\ast$) wegen der Hauptannahme $\phi\circ\phi=\phi$ gilt.
+        Daraus folgt
+
+            \begin{mathe}[mc]{rcccl}
+                v &= &\underbrace{
+                        (v-\phi(v))
+                    }_{\in\ker(\phi)}
+                    +
+                    \underbrace{
+                        \phi(v)
+                    }_{\in\range(\phi)}
+                &\in &\ker(\phi)+\range(\phi).\\
+            \end{mathe}
+
+        Also gelten (i) + (ii), woraus sich die behauptete Gleichung ergibt.
+    \item
+        Da $U_{1}:=\ker(\phi),U_{2}:=\range(\phi)\subseteq V$
+            und damit auch $\ker(\phi)+\range(\phi)\subseteq V$
+        Untervektorräume sind,
+        und da $V$ \uline{endlich dimensional} ist,
+        reicht es auch \textbf{zu zeigen}, dass
+
+            \begin{mathe}[mc]{rcl}
+                \eqtag[eq:A5b:ansatz:2:0]
+                \dim(\ker(\phi)+\range(\phi)) &= &\dim(V).\\
+            \end{mathe}
+
+        Laut der \emph{Dimensionsformel für lineare Abbildungen} gilt:
+
+            \begin{mathe}[mc]{rcl}
+                \eqtag[eq:A5b:ansatz:2:1]
+                \dim(V) &= &\dim(\ker(\phi)) + \dim(\range(\phi)).\\
+            \end{mathe}
+
+        Laut der \emph{Dimensionsformel für Untervektorräume}
+        gilt außerdem:
+
+            \begin{mathe}[mc]{rcl}
+                \eqtag[eq:A5b:ansatz:2:2]
+                \dim(\ker(\phi)) + \dim(\range(\phi))
+                    &= &\dim(U_{1}) + \dim(U_{2})\\
+                    &= &\dim(U_{1} + U_{2}) + \dim(U_{1}\cap U_{2})\\
+                    &= &\dim(\ker(\phi) + \range(\phi)) \\
+                        &&\qquad + \dim(\ker(\phi)\cap \range(\phi))\\
+                    &\textoverset{($\ast$)}{=} &\dim(\ker(\phi) + \range(\phi)) + 0,\\
+            \end{mathe}
+
+        wobei ($\ast$) gilt, weil laut A5 a)
+            $\ker(\phi)\cap \range(\phi)=\{\zerovector\}$
+        ein $0$-dimensionaler Untervektorraum ist.
+        Aus \eqcref{eq:A5b:ansatz:2:1} + \eqcref{eq:A5b:ansatz:2:2}
+        folgt nun \eqcref{eq:A5b:ansatz:2:0}.
+
+        Also stimmt die Dimension des Untervektorraums, $\ker(\phi)+\range(\phi)$,
+        mit der des \uline{endlichdimensionalen} Vektorraums, $V$, überein.
+        Wie oben erzählt folgt daraus: $\ker(\phi)+\range(\phi)=V$.
+        \enndeOfProof
+\end{enumerate}
+
+\begin{punktschema}
+    5   &Argument vollständig (=ausführlich) und logisch gültig.\\
+    \hline
+    4   &Ansatz I: Nichttriviale Inklusion vollständig, aber triviale fehlte.\\
+    \hdashline
+    4   &Ansatz II: Argument vollständig aber eine Lücke, z.\,B. man benutzt die Annahme der Endlichdimensionalität von $V$ nicht.\\
+    \hdashline
+    4   &Ansatz I/II: Argument vollständig kleine Fehler.\\
+    \hline
+    3   &Ein paar Lücken/Fehler, aber generell überzeugend.\\
+    \hdashline
+    3   &Argument baut auf falscher Annahme auf, aber Annahme „rechtfertigt“.\\
+    \hline
+    2   &Nur triviale Inklusion bewiesen.\\
+    \hdashline
+    2   &Argument baut auf falscher Annahme auf und Annahme kaum rechtfertigt.\\
+    \hline
+    1   &Man fing mit ein paar richtigen Schritt von einem Ansatz an aber es fehlte der Rest.\\
+    \hline
+    0   &sonst.\\
+\end{punktschema}
+
+{\footnotesize
+    \textbf{Bemerkungen.}
+    Auf folgende Qualitätsmerkmale wurde geschaut:
+    Wichtig war, dass der ganze Text aus (a) + (b) eine kohärente Erzählung bildete.
+    Bezeichnungen und Ausdrücke mussten sinnvoll sein.
+    Man darf Symbole nicht willkürlich hinschreiben.
+    Es darf keine Kreisschlüsse oder sonstige logische Täuschungen geben.
+    Es darf nicht sein, dass dem Text zu entnehmen ist, dann man sich seiner eigenen Argumentation nicht bewusst ist.
+    In der gesamten Aufgabe musste die Hauptannahme der Aufgabe
+    (in diesem Falle $\phi\circ\phi=\phi$)
+    gebraucht (oder zumindest angesprochen) werden.
+}
+
+\end{document}