From 023d2a2e486a13e46768e9563c7971f8d7eeeb6a Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Sun, 22 Nov 2020 14:14:31 +0100
Subject: [PATCH] master > master: Formatierung + Wortlaut

---
 docs/loesungen.pdf | Bin 393654 -> 393568 bytes
 docs/loesungen.tex |  28 +++++++++++++---------------
 2 files changed, 13 insertions(+), 15 deletions(-)

diff --git a/docs/loesungen.pdf b/docs/loesungen.pdf
index d35ea400350310d4d0ba556b86ee2e56b9d26a29..40bf588d5dc48d7265aad850da81cac89e6aa9d3 100644
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diff --git a/docs/loesungen.tex b/docs/loesungen.tex
index d3397b6..340faa9 100644
--- a/docs/loesungen.tex
+++ b/docs/loesungen.tex
@@ -2003,13 +2003,12 @@ Mit diesem Mittel können wir nun die Hauptaussage in der Aufgabe formulieren:
         für $i\in\{1,2,\ldots,m\}$.
         Da ${\mathbf{z}^{(1)},\mathbf{z}^{(2)},\ldots,\mathbf{z}^{(m)}\in\reell^{n}}$,
         können wir eine \emph{maximale Menge} ${I_{0}\subseteq\{1,2,\ldots,m\}}$ finden,
-        so dass $(\mathbf{z}^{(i)})_{i\in I_{0}}$ linear unabhängige Vektoren sind.
-        Aus der Maximalität folgt, dass für jedes ${k\in\{1,2,\ldots,m\}\ohne I_{0}}$
-        $(\mathbf{z}^{(i)})_{i\in I_{0}\cup\{k\}}$ \emph{linear abhängig} sind.
+        so dass $(\mathbf{z}^{(i)})_{i\in I_{0}}$ aus linear unabhängigen Vektoren besteht.
         Wegen der Dimension von $\reell^{n}$ gilt ${|I|\leq\min\{m,n\}=n}$.
-        Aus der linearer Unabhängigkeit von den $(\mathbf{z}^{(i)})_{i\in I_{0}}$ folgt,
-        dass es (eindeutige) Koeffizienten $c_{k,i}\in\reell$ für $i\in I_{0}$ gibt,
-        so dass
+        Sei ${k\in\{1,2,\ldots,m\}\ohne I_{0}}$ beliebig.
+        Wegen Maximalität muss $(\mathbf{z}^{(i)})_{i\in I_{0}\cup\{k\}}$ \emph{linear abhängig} sein.
+        Und wegen der linearen Unabhängigkeit von $(\mathbf{z}^{(i)})_{i\in I_{0}}$
+        existieren (eindeutige) Koeffizienten $c_{k,i}\in\reell$ für $i\in I_{0}$ so dass
 
             \begin{mathe}[mc]{rcl}
                 \eqtag[eq:1:\beweislabel]
@@ -3678,8 +3677,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
 
         \begin{einzug}[\rtab][\rtab]
         \begin{proof}
-            Wir zeigen \Cref{\beweislabel} per Induktion.
-            Als Induktionsanfang widmen wir uns den Fällen $n\leq 2$.
+            Wir zeigen dies per Induktion mit den Fällen $n\leq 2$ als Induktionsanfang.
 
             \begin{kompaktenum}[\rtab][\rtab]
                 \item[\uwave{{\bfseries Induktionsanfang:}}]
@@ -3693,7 +3691,9 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
 
                     Also gilt $\Phi(1)$.
                 \item[]
-                    Sei $n=2$. Dann gilt für alle endlichen Mengen $E_{1},E_{2}$
+                    Sei $n=2$.
+                    Laut \Cref{lemm:1:ska:4:ex:10} (siehe unten)
+                    gilt für alle endlichen Mengen $E_{1},E_{2}$
 
                         \begin{mathe}[mc]{rcccccl}
                             |\prod_{i=1}^{2}E_{i}|
@@ -3702,7 +3702,6 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
                             &= &\prod_{i=1}^{2}|E_{i}|.\\
                         \end{mathe}
 
-                    (Dieses Resultat haben wir in \Cref{lemm:1:ska:4:ex:10} ausgelagert.)\\
                     Also gilt $\Phi(2)$.
 
                 \item[\uwave{{\bfseries Induktionsvoraussetzung:}}]
@@ -3741,7 +3740,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
 
         \begin{einzug}[\rtab][\rtab]
         \begin{proof}
-            Wir beweisen dies per Induktion über $|Y|$ durchführen.
+            Wir zeigen dies per Induktion über $|Y|$ mit den Fällen $|Y|\leq 1$ als Induktionsanfang.
 
                 \begin{kompaktenum}[\rtab][\rtab]
                     \item[\uwave{{\bfseries Induktionsanfang:}}]
@@ -3769,12 +3768,11 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
                     \item[\uwave{{\bfseries Induktionsvoraussetzung:}}]
                         Sei $n>1$.
                         Angenommen, $|X\times Y'|=|X|\cdot |Y'|$
-                        für alle $k$-elementigen Mengen, $Y'$
-                        und für alle $k<n$.
+                        für alle $n-1$-elementigen Mengen, $Y'$.
 
                     \item[\uwave{{\bfseries Induktionsschritt:}}]
                         Sei $Y$ eine beliebige $n$-elementige Menge.\\
-                        \textbf{Zu zeigen:} $|X\times Y|=|X|\times|Y|$ gilt.\\
+                        \textbf{Zu zeigen:} $|X\times Y|=|X|\cdot|Y|$ gilt.\\
                         Da $n>0$, können wir ein beliebiges $y_{0}\in Y$ fixieren.\\
                         Setze $Y':=Y\ohne\{y_{0}\}$.
                         Da $Y'$ $n-1$-elementig ist, gilt per Induktionsvoraussetzung
@@ -3876,7 +3874,7 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
                 Da $X_{0}$ $n$-elementig ist und $x_{0}\in X_{0}$ und $G(x_{0})$,
                 gilt per IV (\textdagger)~$\forall{x\in X_{0}:~}G(x)$.
 
-                Wähle nun irgendeinen der Fische, $\tilde{x}\in X_{0}$ und setze $X':=X\ohne\{\tilde{x}\}$.
+                Wähle nun irgendeinen der Fische, $\tilde{x}\in X_{0}$ und setze $X':=X\ohne\{\tilde{x}\}$.\\
                 O.\,E. können wir $\tilde{x}:=x_{0}$ wählen, sodass $X'=X_{1}$ gilt.\\
                 Die Teilmenge $X_{1}$ ist nun eine $n$-elementige Menge mit mindestens $n-1$ Goldfischen.\\
                 \fbox{Also $\exists{x\in X_{1}:~}G(x)$.}\\