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Micron Document

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<title>Neunerlemma</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Neunerlemma</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Das Neunerlemma, wegen der Struktur des unten abgebildeten Diagramms auch 3×3-Lemma genannt, ist eine <a href="Mathematik" title="Mathematik">mathematische</a> Aussage über <a href="Kommutatives_Diagramm" title="Kommutatives Diagramm">kommutierende Diagramme</a> und <a href="Exakte_Folge" class="mw-redirect" title="Exakte Folge">exakte Folgen</a>, die sowohl für jede <a href="Abelsche_Kategorie" title="Abelsche Kategorie">abelsche Kategorie</a> als auch für die Kategorie der <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a> gültig ist.

<div class="mw-heading mw-heading2"><h2 id="Aussage">Aussage</h2></div>

Ist (in einer abelschen Kategorie oder der Kategorie der Gruppen) das Diagramm

kommutativ und sind alle Spalten sowie die unteren beiden Zeilen exakt, so ist auch die obere Zeile exakt.

Ebenso gilt: Sind alle Spalten sowie die oberen beiden Zeilen exakt, so ist auch die untere Zeile exakt.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Beweis">Beweis</h2></div>

Der Beweis erfolgt durch <a href="Diagrammjagd" class="mw-redirect" title="Diagrammjagd">Diagrammjagd</a>, zunächst unter der Annahme, dass das Diagramm die Kategorie der Gruppen betrifft.

Der Einfachheit halber seien alle horizontalen Abbildungen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, alle vertikalen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> bezeichnet.

Das neutrale Element der Gruppen heiße jeweils <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

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<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">.

Der Beweis zeigt die typische Eigenschaft von Diagrammjagden, dass der schriftliche Beweis zwar aus lauter trivialen Einzelschritten besteht, die zusammen jedoch verwirrend oder unmotiviert wirken – erst wenn man die Schritte am Diagramm nachverfolgt, werden die Zusammenhänge einleuchtend.

Seien zunächst alle Spalten sowie die unteren beiden Zeilen exakt.

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<annotation encoding="application/x-tex">{\displaystyle a_{1}\in A_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9cec5236c3534d951e94dc45ba78edc3fa72f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{1}\in A_{1}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(a_{1})=e}">

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(a_{1})=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/108a84ac77435a366454388fbe40e070b293d588.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.614ex; height:2.843ex;" alt="{\displaystyle h(a_{1})=e}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(a_{1}))=v(h(a_{1}))=v(e)=e}">

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<annotation encoding="application/x-tex">{\displaystyle h(v(a_{1}))=v(h(a_{1}))=v(e)=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b28cbd2fb65f1ba7417bdc6ec2d70dcd0b7bad26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.138ex; height:2.843ex;" alt="{\displaystyle h(v(a_{1}))=v(h(a_{1}))=v(e)=e}" loading="lazy">. Hieraus folgt mit der <a href="Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">Injektivität</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon A_{2}\to B_{2}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h\colon A_{2}\to B_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd899fc01c51edbaa762d76338e4e29dae9a7643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.603ex; height:2.509ex;" alt="{\displaystyle h\colon A_{2}\to B_{2}}" loading="lazy"> auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(a_{1})=e}">

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<annotation encoding="application/x-tex">{\displaystyle v(a_{1})=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3819cf2954eca2eb46e2c885ad75ed31d6d35f87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.403ex; height:2.843ex;" alt="{\displaystyle v(a_{1})=e}" loading="lazy"> und mit der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\colon A_{1}\to A_{2}}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle v\colon A_{1}\to A_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c1a7b63d587b9fe6904705d9a6522bf7e573504.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.37ex; height:2.509ex;" alt="{\displaystyle v\colon A_{1}\to A_{2}}" loading="lazy"> schließlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=e}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a_{1}=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/503bd5e2367080a6cb0fcf69f4102df683a18899.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.466ex; height:2.009ex;" alt="{\displaystyle a_{1}=e}" loading="lazy">.</li>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a_{1}\in A_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9cec5236c3534d951e94dc45ba78edc3fa72f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{1}\in A_{1}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(h(h(a_{1})))=h(h(v(a_{1})))=e}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle v(h(h(a_{1})))=h(h(v(a_{1})))=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a9ba87dca20f788ee85296e18b05e23b6725007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.315ex; height:2.843ex;" alt="{\displaystyle v(h(h(a_{1})))=h(h(v(a_{1})))=e}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(h(a_{1}))=e}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(h(a_{1}))=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63d85c5448af440cb1248b9c98b38d2786af3d95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:2.843ex;" alt="{\displaystyle h(h(a_{1}))=e}" loading="lazy">.</li>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle b_{1}\in B_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/959c44c38e9b843bf7f7da115f8bee260af9c64d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{1}\in B_{1}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(b_{1})=e}">

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(b_{1})=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64707b388e515fb2088fea7b9139c2a0e4d67af2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.382ex; height:2.843ex;" alt="{\displaystyle h(b_{1})=e}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(b_{1}))=v(h(b_{1}))=e}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(v(b_{1}))=v(h(b_{1}))=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8cf77cfc1a3174b6eb864b4e62c30cab4a5d4d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.554ex; height:2.843ex;" alt="{\displaystyle h(v(b_{1}))=v(h(b_{1}))=e}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(b_{1})=h(a_{2})}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(b_{1})=h(a_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/caf865f6e0b6433bddf6590b52fedfb743ed6344.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.519ex; height:2.843ex;" alt="{\displaystyle v(b_{1})=h(a_{2})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}\in A_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{2}\in A_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1566d4c163caf81ec812aa7946af5bf35cd92b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{2}\in A_{2}}" loading="lazy">. Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(a_{2}))=v(h(a_{2}))=v(v(b_{1}))=e}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(v(a_{2}))=v(h(a_{2}))=v(v(b_{1}))=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a032bde8f2c244447d8c997aeb8853b601e6bbec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.043ex; height:2.843ex;" alt="{\displaystyle h(v(a_{2}))=v(h(a_{2}))=v(v(b_{1}))=e}" loading="lazy"> folgt auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(a_{2})=e}">

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<annotation encoding="application/x-tex">{\displaystyle v(a_{2})=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f63d8ac5b7bde1b4c5b2a393f7ce2e8c320ef359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.403ex; height:2.843ex;" alt="{\displaystyle v(a_{2})=e}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}=v(a_{1})}">

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a_{2}=v(a_{1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617176facfa78946ef438e94f5e33bcb7fd85674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle a_{2}=v(a_{1})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\in A_{1}}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a_{1}\in A_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9cec5236c3534d951e94dc45ba78edc3fa72f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{1}\in A_{1}}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(h(a_{1}))=h(v(a_{1}))=h(a_{2})=v(b_{1})}">

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<annotation encoding="application/x-tex">{\displaystyle v(h(a_{1}))=h(v(a_{1}))=h(a_{2})=v(b_{1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28c95206d040e7252df3c3bbac5856e1930137c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.455ex; height:2.843ex;" alt="{\displaystyle v(h(a_{1}))=h(v(a_{1}))=h(a_{2})=v(b_{1})}" loading="lazy">, woraus bereits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}=h(a_{1})}">

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<annotation encoding="application/x-tex">{\displaystyle b_{1}=h(a_{1})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle c_{1}\in C_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cafd86f57cdd9741d4ba5811dcf6214e45df0385.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.618ex; height:2.509ex;" alt="{\displaystyle c_{1}\in C_{1}}" loading="lazy">, so gibt es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}\in B_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{2}\in B_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5044f24b3d5495ceed7592fdf57bf53da9b4d3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{2}\in B_{2}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(b_{2})=v(c_{1})}">

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(b_{2})=v(c_{1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5837c63a9e72ef36ade5fe6f030a1124f63beafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.297ex; height:2.843ex;" alt="{\displaystyle h(b_{2})=v(c_{1})}" loading="lazy">. Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(b_{2}))=v(h(b_{2}))=v(v(c_{1}))=e}">

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<annotation encoding="application/x-tex">{\displaystyle h(v(b_{2}))=v(h(b_{2}))=v(v(c_{1}))=e}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4efe1ad8229274c467bff92222ea7d17b173451d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.588ex; height:2.843ex;" alt="{\displaystyle h(v(b_{2}))=v(h(b_{2}))=v(v(c_{1}))=e}" loading="lazy"> gibt es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}\in A_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{3}\in A_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/464166c6f36f9036cc80d163e622c419a5cac000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{3}\in A_{3}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(a_{3})=v(b_{2})}">

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</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h(a_{3})=v(b_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fbc632993df3180ddb09225f44cd9dd88a4fe0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.519ex; height:2.843ex;" alt="{\displaystyle h(a_{3})=v(b_{2})}" loading="lazy">. Weiter gibt es ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}\in A_{2}}">

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</mrow>

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{2}\in A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1566d4c163caf81ec812aa7946af5bf35cd92b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{2}\in A_{2}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(a_{2})=a_{3}}">

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</mrow>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(a_{2})=a_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d66662720808f59a18de41785e9e1efab81d75a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle v(a_{2})=a_{3}}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(h(a_{2}))=h(v(a_{2}))=h(a_{3})=v(b_{2})}">

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</mrow>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(h(a_{2}))=h(v(a_{2}))=h(a_{3})=v(b_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eae82926d4236df2e0767969e549074a0c5d7084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.455ex; height:2.843ex;" alt="{\displaystyle v(h(a_{2}))=h(v(a_{2}))=h(a_{3})=v(b_{2})}" loading="lazy">. Somit unterscheiden sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(a_{2})}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(a_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0644cd27ef18e4f04081e24ffb7c201a6c64e30e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.432ex; height:2.843ex;" alt="{\displaystyle h(a_{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2530a260ad35bf21ee61f1f4d6493ae0474f6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{2}}" loading="lazy"> um <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(b_{1})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(b_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/807aea08e67dd028b25f327b543ab2945c9eecf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.989ex; height:2.843ex;" alt="{\displaystyle v(b_{1})}" loading="lazy"> für ein geeignetes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}\in B_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}\in B_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/959c44c38e9b843bf7f7da115f8bee260af9c64d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{1}\in B_{1}}" loading="lazy">, d. h. es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}=v(b_{1})\cdot h(a_{2})}">

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</mrow>

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</mrow>

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}=v(b_{1})\cdot h(a_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e64e551a264d03b00b8c4a396aa7d4df53da30bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.25ex; height:2.843ex;" alt="{\displaystyle b_{2}=v(b_{1})\cdot h(a_{2})}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(c_{1})=h(b_{2})=h(v(b_{1})\cdot h(a_{2}))=h(v(b_{1}))\cdot h(h(a_{2}))=h(v(b_{1}))=v(h(b_{1}))}">

<msub>

</mrow>

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</mrow>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(c_{1})=h(b_{2})=h(v(b_{1})\cdot h(a_{2}))=h(v(b_{1}))\cdot h(h(a_{2}))=h(v(b_{1}))=v(h(b_{1}))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cc3dd5212c846b6eab76426a36d9a1596ee8924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.609ex; height:2.843ex;" alt="{\displaystyle v(c_{1})=h(b_{2})=h(v(b_{1})\cdot h(a_{2}))=h(v(b_{1}))\cdot h(h(a_{2}))=h(v(b_{1}))=v(h(b_{1}))}" loading="lazy"> und schließlich auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}=h(b_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}=h(b_{1})}</annotation>

</semantics>

Alle Punkte zusammen zeigen die Exaktheit der ersten Zeile.

Seien jetzt alle Spalten sowie die oberen beiden Zeilen exakt.

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{3}\in C_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/362c83164fc74a22c39df3f183f688dbc1fa824e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.618ex; height:2.509ex;" alt="{\displaystyle c_{3}\in C_{3}}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}=v(c_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{3}=v(c_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0c9c4cc0a34712a27500ca0f3ad885eda98dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.158ex; height:2.843ex;" alt="{\displaystyle c_{3}=v(c_{2})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}\in C_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{2}\in C_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d39cfc90ce4d27bf210a9b617302b500f977cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.618ex; height:2.509ex;" alt="{\displaystyle c_{2}\in C_{2}}" loading="lazy"> und dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}=h(b_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{2}=h(b_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/381b350bf5d53d1a62b592e3172c9b16fd5913cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.36ex; height:2.843ex;" alt="{\displaystyle c_{2}=h(b_{2})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}\in B_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}\in B_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5044f24b3d5495ceed7592fdf57bf53da9b4d3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{2}\in B_{2}}" loading="lazy">, jeweils per <a href="Surjektivit%C3%A4t" class="mw-redirect" title="Surjektivität">Surjektivität</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\colon C_{2}\to C_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\colon C_{2}\to C_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eb191e385e0bcd28d03f96838999de75b5a7d91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.208ex; height:2.509ex;" alt="{\displaystyle v\colon C_{2}\to C_{3}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon B_{2}\to C_{2}}">

<msub>

</mrow>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\colon B_{2}\to C_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f8cdb8910d5206e198986289ae8e9eb32cc28c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.521ex; height:2.509ex;" alt="{\displaystyle h\colon B_{2}\to C_{2}}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(b_{2}))=v(h(b_{2}))=c_{3}}">

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</mrow>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v(b_{2}))=v(h(b_{2}))=c_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c82a861a135a7413389fb314a62d48eeb1bc5dea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.532ex; height:2.843ex;" alt="{\displaystyle h(v(b_{2}))=v(h(b_{2}))=c_{3}}" loading="lazy">.</li>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{3}\in A_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/464166c6f36f9036cc80d163e622c419a5cac000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{3}\in A_{3}}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}=v(a_{2})}">

<msub>

</mrow>

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</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{3}=v(a_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08f0937e40c48d2f58a7d62085323a05b2c99962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle a_{3}=v(a_{2})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}\in A_{2}}">

<msub>

</mrow>

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{2}\in A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1566d4c163caf81ec812aa7946af5bf35cd92b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{2}\in A_{2}}" loading="lazy">. Dann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(h(a_{3}))=h(h(v(a_{2})))=v(h(h(a_{2})))=v(e)=e}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(h(a_{3}))=h(h(v(a_{2})))=v(h(h(a_{2})))=v(e)=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dbd543947de18c9f5f509a2a0a99339682de27a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.113ex; height:2.843ex;" alt="{\displaystyle h(h(a_{3}))=h(h(v(a_{2})))=v(h(h(a_{2})))=v(e)=e}" loading="lazy">.</li>

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{3}\in B_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf4002ba19bd2040c173ee3d0e28b9f2d738c6f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{3}\in B_{3}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(b_{3})=e}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(b_{3})=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98b696427d8b2043c4d1a2f01dfc87e6da957219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.382ex; height:2.843ex;" alt="{\displaystyle h(b_{3})=e}" loading="lazy"> und wählen wir ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}\in B_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}\in B_{2}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(b_{2})=b_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c09572199a78a2a49afe3db4e369758e6e754382.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.139ex; height:2.843ex;" alt="{\displaystyle v(b_{2})=b_{3}}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(h(b_{2}))=h(v(b_{2}))=h(b_{3})=e}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(h(b_{2}))=h(v(b_{2}))=h(b_{3})=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c73ecdd2b11f20cc93bf24cb8c1f06a4e76a34f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.853ex; height:2.843ex;" alt="{\displaystyle v(h(b_{2}))=h(v(b_{2}))=h(b_{3})=e}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(b_{2})=v(c_{1})}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(b_{2})=v(c_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5837c63a9e72ef36ade5fe6f030a1124f63beafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.297ex; height:2.843ex;" alt="{\displaystyle h(b_{2})=v(c_{1})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}\in C_{1}}">

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</mrow>

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<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}\in C_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cafd86f57cdd9741d4ba5811dcf6214e45df0385.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.618ex; height:2.509ex;" alt="{\displaystyle c_{1}\in C_{1}}" loading="lazy">. Weiter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}=h(b_{1})}">

<msub>

</mrow>

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<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}=h(b_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be5d2d5e453ea170118884cd19b9d9b8d27f4ced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.36ex; height:2.843ex;" alt="{\displaystyle c_{1}=h(b_{1})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}\in B_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}\in B_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/959c44c38e9b843bf7f7da115f8bee260af9c64d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{1}\in B_{1}}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(b_{1}))=v(h(b_{1}))=v(c_{1})=h(b_{2})}">

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v(b_{1}))=v(h(b_{1}))=v(c_{1})=h(b_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b5824c9e17f4d123081ed8cd349f86de0b67442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.767ex; height:2.843ex;" alt="{\displaystyle h(v(b_{1}))=v(h(b_{1}))=v(c_{1})=h(b_{2})}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}=v(b_{1})\cdot h(a_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}=v(b_{1})\cdot h(a_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e64e551a264d03b00b8c4a396aa7d4df53da30bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.25ex; height:2.843ex;" alt="{\displaystyle b_{2}=v(b_{1})\cdot h(a_{2})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}\in A_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{2}\in A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1566d4c163caf81ec812aa7946af5bf35cd92b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{2}\in A_{2}}" loading="lazy">. Schließlich ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(a_{2}))=v(h(a_{2}))=v(v(b_{1}))\cdot v(h(a_{2}))=v(v(b_{1})\cdot h(a_{2}))=v(b_{2})=b_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v(a_{2}))=v(h(a_{2}))=v(v(b_{1}))\cdot v(h(a_{2}))=v(v(b_{1})\cdot h(a_{2}))=v(b_{2})=b_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b51535fde36c7d3b7bfb0dff638ba90d0728df51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.282ex; height:2.843ex;" alt="{\displaystyle h(v(a_{2}))=v(h(a_{2}))=v(v(b_{1}))\cdot v(h(a_{2}))=v(v(b_{1})\cdot h(a_{2}))=v(b_{2})=b_{3}}" loading="lazy">.</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{3}\in A_{3}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(a_{3})=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d18100419926d3d00575b840c88fb15a9704694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.614ex; height:2.843ex;" alt="{\displaystyle h(a_{3})=e}" loading="lazy"> und wählen wir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}\in A_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{2}\in A_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1566d4c163caf81ec812aa7946af5bf35cd92b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{2}\in A_{2}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(a_{2})=a_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(a_{2})=a_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d66662720808f59a18de41785e9e1efab81d75a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle v(a_{2})=a_{3}}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(h(a_{2}))=h(v(a_{2}))=h(a_{3})=e}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(h(a_{2}))=h(v(a_{2}))=h(a_{3})=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/023804789e38fcc7920ca81c5ba73b52143ddc71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.55ex; height:2.843ex;" alt="{\displaystyle v(h(a_{2}))=h(v(a_{2}))=h(a_{3})=e}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(a_{2})=v(b_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(a_{2})=v(b_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d39801be64a27bfba9aebd7f96edc979a5933191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.519ex; height:2.843ex;" alt="{\displaystyle h(a_{2})=v(b_{1})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}\in B_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}\in B_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/959c44c38e9b843bf7f7da115f8bee260af9c64d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.711ex; height:2.509ex;" alt="{\displaystyle b_{1}\in B_{1}}" loading="lazy">. Es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(h(b_{1}))=h(v(b_{1}))=h(h(a_{2}))=e}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(h(b_{1}))=h(v(b_{1}))=h(h(a_{2}))=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ca717c9a83d0940ccf05a09d8293cc889a097c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.233ex; height:2.843ex;" alt="{\displaystyle v(h(b_{1}))=h(v(b_{1}))=h(h(a_{2}))=e}" loading="lazy">, daher bereits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(b_{1})=e}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(b_{1})=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64707b388e515fb2088fea7b9139c2a0e4d67af2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.382ex; height:2.843ex;" alt="{\displaystyle h(b_{1})=e}" loading="lazy">. Folglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}=h(a_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}=h(a_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5db40e838f3aee127391b852cc3c8b3b5d4b14cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.583ex; height:2.843ex;" alt="{\displaystyle b_{1}=h(a_{1})}" loading="lazy"> für ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\in A_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1}\in A_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9cec5236c3534d951e94dc45ba78edc3fa72f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle a_{1}\in A_{1}}" loading="lazy">. Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v(a_{1}))=v(h(a_{1}))=v(b_{1})=h(a_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v(a_{1}))=v(h(a_{1}))=v(b_{1})=h(a_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74790f7beed5682f220d35863d32c30e2b7662af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.455ex; height:2.843ex;" alt="{\displaystyle h(v(a_{1}))=v(h(a_{1}))=v(b_{1})=h(a_{2})}" loading="lazy"> folgt bereits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}=v(a_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{2}=v(a_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/617176facfa78946ef438e94f5e33bcb7fd85674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle a_{2}=v(a_{1})}" loading="lazy"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}=v(a_{2})=v(v(a_{1}))=e}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{3}=v(a_{2})=v(v(a_{1}))=e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97c4812f9877a953d0814d162914aa7e58a50371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.042ex; height:2.843ex;" alt="{\displaystyle a_{3}=v(a_{2})=v(v(a_{1}))=e}" loading="lazy">.</li></ul>

Zusammen ergibt dies wiederum die Exaktheit der letzten Zeile.

Der zunächst für Gruppen durchgeführte Beweis gilt (ggf. in additive Schreibweise übersetzt) ebenso für <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppen</a> oder auch für <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> über einem <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</a>.

Durch den <a href="Einbettungssatz_von_Mitchell" title="Einbettungssatz von Mitchell">Einbettungssatz von Mitchell</a> ist dies aber bereits ausreichend, um das Neunerlemma für alle abelschen Kategorien zu beweisen.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="F%C3%BCnferlemma" title="Fünferlemma">Fünferlemma</a></li>

<li><a href="Schlangenlemma" title="Schlangenlemma">Schlangenlemma</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a>: Homology, Springer <a href="Grundlehren_der_mathematischen_Wissenschaften" title="Grundlehren der mathematischen Wissenschaften">Grundlehren der mathematischen Wissenschaften</a> Band 114 (1967), Kapitel II, Lemma 5.1 (The 3x3-Lemma)

</li>

</ol></div><div><div class="zim-footer">

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