English+GermanWikipedia - /page/entry.mu

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<title>Hilfssatz</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hilfssatz</span></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"><p>Ein <b>Hilfssatz</b> oder <b>Lemma</b> (<span style="font-style:normal;font-weight:normal"><a href="Altgriechische_Sprache" title="Altgriechische Sprache">altgriechisch</a></span> <span lang="grc-Grek" class="Grek" style="font-style:normal">λῆμμα</span> <style data-mw-deduplicate="TemplateStyles:r261937631">

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</style><span class="Latn" lang="grc-Latn" style="font-weight:normal;font-style:italic">lēmma</span> ‚Einnahme‘, ‚Annahme‘; <a href="Plural" title="Plural">Plural</a>: „Lemmata“)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> ist eine <a href="Mathematik" title="Mathematik">mathematische</a> oder <a href="Logik" title="Logik">logische</a> Aussage, die im Beweis eines <a href="Satz_(Mathematik)" title="Satz (Mathematik)">Satzes</a> verwendet wird, der aber selbst nicht der Rang eines Satzes eingeräumt wird. Die Unterscheidung von Sätzen und Lemmata ist fließend und nicht objektiv. Der Begriff „Lemma“ lässt sich auch mit „Stichwort“ oder auch „Hauptgedanke“ übersetzen. Dies signalisiert, dass es sich um einen Schlüsselgedanken handelt, der in vielen Situationen nützlich ist.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Berühmte_Lemmata"><span id="Ber.C3.BChmte_Lemmata"></span>Berühmte Lemmata</h3></div>

<p>Lemmata tragen häufig die Namen ihres Entdeckers. Beispiele hierfür sind:

</p>

<ul><li><a href="Lemma_von_Zorn" title="Lemma von Zorn">Lemma von Zorn</a></li>

<li><a href="Lemma_von_Sperner" title="Lemma von Sperner">Lemma von Sperner</a></li>

<li><a href="Lemma_von_Euklid" title="Lemma von Euklid">Lemma von Euklid</a></li>

<li><a href="Lemma_von_Gau%C3%9F" class="mw-redirect" title="Lemma von Gauß">Lemma von Gauß</a></li>

<li><a href="Reihe_(Gruppentheorie)#Lemma_von_Zassenhaus" title="Reihe (Gruppentheorie)">Lemma von Zassenhaus</a> (auch <i>Schmetterlingslemma</i>) über subnormale Reihen von Gruppen</li>

<li><a href="Lemma_von_Margulis#Lemma_von_Margulis_(gruppentheoretische_Formulierung)" title="Lemma von Margulis">Lemma von Margulis</a> (häufiger <i>Lemma von Margulis-Zassenhaus</i> oder auch <i>Lemma von Kazhdan-Margulis-Zassenhaus</i>) über Matrizengruppen</li></ul>

<p>Weitere Beispiele finden sich in der <a href="Liste_mathematischer_S%C3%A4tze" title="Liste mathematischer Sätze">Liste mathematischer Sätze</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_für_die_Nutzung_eines_Lemmas"><span id="Beispiel_f.C3.BCr_die_Nutzung_eines_Lemmas"></span>Beispiel für die Nutzung eines Lemmas</h3></div>

<p>Man kann beispielsweise zeigen, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">

<msqrt>

</msqrt>

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</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy"></span> <a href="Irrationale_Zahl" title="Irrationale Zahl">irrational</a> ist (als Satz), wenn man voraussetzen kann, dass <a href="Quadrat_(Arithmetik)" class="mw-redirect" title="Quadrat (Arithmetik)">Quadrate</a> gerader Zahlen wieder gerade sind, Quadrate ungerader Zahlen jedoch stets ungerade Zahlen ergeben (diese Aussage entspräche dem Lemma). Um strukturierter vorzugehen, beweist man die beiden Tatsachen einzeln, wobei die Tatsache des Hilfssatzes (des Lemmas) später auf weitere Fälle oder Beweise angewendet werden kann, wohingegen der „Satz“ eine spezielle Aussage liefert.

</p><p>Um das vorangegangene Beispiel umzusetzen, ginge man (zum Beispiel in einer <a href="Vorlesung" title="Vorlesung">Vorlesung</a>) folgendermaßen vor.

</p><p><b>Lemma:</b> Quadrate gerader und ungerader ganzer Zahlen sind stets gerade bzw. ungerade.

</p><p><b>Beweis:</b> Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Z} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Z} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fff96cba8be0fa52dd0b472c9f40e7d6a3cb75db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {Z} }" loading="lazy"></span> vorgegeben. Zu zeigen ist, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf0bf28fd28f45d07e1ceb909ce333c18c558c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}" loading="lazy"></span> der entsprechenden Behauptung genügt, d. h. wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2y}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5e265b08f5a3b1142c21a51ff6e06004c35ddf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle x=2y}" loading="lazy"></span> (gerade) bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2y+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2y+1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8470a4f3368ae6a38b95edcf6bfd427b16f185b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle x=2y+1}" loading="lazy"></span> (ungerade) für ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {Z} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7e6f12a950fae74d6a37b86f7a4bca9174475e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.546ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {Z} }" loading="lazy"></span> ist, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf0bf28fd28f45d07e1ceb909ce333c18c558c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}" loading="lazy"></span> gerade bzw. ungerade.

</p><p>Beide Fälle werden separat behandelt. Im ersten Fall (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2y}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5e265b08f5a3b1142c21a51ff6e06004c35ddf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle x=2y}" loading="lazy"></span>) hat man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=(2y)^{2}=2^{2}\cdot y^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}=(2y)^{2}=2^{2}\cdot y^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c61d9ec0ef2ee97b4d28e1e8c0801093aadea678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.873ex; height:3.176ex;" alt="{\displaystyle x^{2}=(2y)^{2}=2^{2}\cdot y^{2}}" loading="lazy"></span> (gemäß den <a href="Potenz_(Mathematik)#Potenzgesetze" title="Potenz (Mathematik)">Potenzrechenregeln</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =2\cdot 2y^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30489909d12c850a38ef9ac914650bd54f053f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.672ex; height:3.009ex;" alt="{\displaystyle =2\cdot 2y^{2}}" loading="lazy"></span>, also eine gerade Zahl. Im anderen Fall (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2y+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2y+1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8470a4f3368ae6a38b95edcf6bfd427b16f185b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle x=2y+1}" loading="lazy"></span>) ergibt sich <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=(2y+1)^{2}=(2y)^{2}+2\cdot 2y\cdot 1+1^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}=(2y+1)^{2}=(2y)^{2}+2\cdot 2y\cdot 1+1^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3960081a70788f7584249243983daec90220c89e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.845ex; height:3.176ex;" alt="{\displaystyle x^{2}=(2y+1)^{2}=(2y)^{2}+2\cdot 2y\cdot 1+1^{2}}" loading="lazy"></span> (nach <a href="Binomische_Formel" class="mw-redirect" title="Binomische Formel">Binomischer Formel</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =2\cdot (2y^{2}+2y)+1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =2\cdot (2y^{2}+2y)+1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbe8c6c3bed68448a348032884e6ec59a713be1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.643ex; height:3.176ex;" alt="{\displaystyle =2\cdot (2y^{2}+2y)+1}" loading="lazy"></span>, also eine ungerade Zahl.

</p><p><b>Satz:</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy"></span> ist irrational, also gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\in \mathbb {R} \setminus \mathbb {Q} }">

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\in \mathbb {R} \setminus \mathbb {Q} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aabefc722c665f769adfa05d66df68de6f2f3c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.62ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2}}\in \mathbb {R} \setminus \mathbb {Q} }" loading="lazy"></span>.

</p><p><b>Beweis:</b> Die behauptete Aussage wird bewiesen, indem die Annahme, das Gegenteil sei richtig, zum Widerspruch geführt wird (Widerspruchsbeweis).

</p><p>Es wird angenommen, es gelte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\in \mathbb {Q} }">

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\in \mathbb {Q} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18cfedc409223ea8ba4e5e3da02bdec112f42e9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.747ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\in \mathbb {Q} }" loading="lazy"></span>. Dann gibt es zueinander <a href="Teilerfremdheit" title="Teilerfremdheit">teilerfremde</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/175ebd03e0644b0967a63d648c2843a5e883257b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.621ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {Z} }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b\in \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b2dd48f69fc66310b6f437f9f0de49bfb7d787.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.516ex; height:2.176ex;" alt="{\displaystyle b\in \mathbb {N} }" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}={\frac {a}{b}}}">

<msqrt>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}={\frac {a}{b}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90480fe7df7a77399cc1ff591e7d379cd16cc260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.263ex; height:4.843ex;" alt="{\displaystyle {\sqrt {2}}={\frac {a}{b}}}" loading="lazy"></span>. Quadriert man diese Gleichung und multipliziert beide Seiten mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acf98b04bfc723606ebb4a7942fa3ab94becd2ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.052ex; height:2.676ex;" alt="{\displaystyle b^{2}}" loading="lazy"></span>, erhält man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot b^{2}=a^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\cdot b^{2}=a^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df50f46c772a666fb36f2a8e69368b5ae4873f7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.276ex; height:2.676ex;" alt="{\displaystyle 2\cdot b^{2}=a^{2}}" loading="lazy"></span>. Weil die linke Seite gerade ist, ist auch die rechte gerade. Nach dem vorausgegangenen Lemma ist dann auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> gerade (denn wäre <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> ungerade, wäre <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f564e5dc0b6e68af32ca8614e972f5b36e944a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{2}}" loading="lazy"></span> ungerade) und es gibt ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8d2be34156860619e6114c8fde955043a405af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {Z} }" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b68503ad5671b4ba2bf45842d056541690d749e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.176ex;" alt="{\displaystyle a=2c}" loading="lazy"></span>. Aus der Gleichung folgt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}={\frac {a^{2}}{2}}={\frac {(2c)^{2}}{2}}=2c^{2}}">

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{2}={\frac {a^{2}}{2}}={\frac {(2c)^{2}}{2}}=2c^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f09142fc901d8f26e24dd6a5d0f4cac01375c6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.56ex; height:5.843ex;" alt="{\displaystyle b^{2}={\frac {a^{2}}{2}}={\frac {(2c)^{2}}{2}}=2c^{2}}" loading="lazy"></span>, woraus man erkennt, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acf98b04bfc723606ebb4a7942fa3ab94becd2ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.052ex; height:2.676ex;" alt="{\displaystyle b^{2}}" loading="lazy"></span> und damit auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> (wieder wegen des Lemmas) gerade sind. Dies widerspricht der Annahme, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> teilerfremd gewählt worden sind. Damit ist die Annahme, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy"></span> sei rational, falsch und der Satz ist bewiesen.

</p><p>Beim Beweis wurde zweimal das vorausgehende Lemma benutzt.

</p>

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

<ul><li><a href="Doron_Zeilberger" title="Doron Zeilberger">Doron Zeilberger</a>, <a rel="nofollow" class="external text" href="https://sites.math.rutgers.edu/~zeilberg/Opinion82.html">Opinion 82: A Good Lemma is Worth a Thousand Theorems</a> (englisch)</li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Wilhelm Pape: <i>Handwörterbuch der griechischen Sprache</i>. Braunschweig, 1914, Band 2, S. 39, Stichwort λῆμμα. (<a rel="nofollow" class="external text" href="http://www.zeno.org/Pape-1880/A/%CE%BB%E1%BF%86%CE%BC%CE%BC%CE%B1">bei zeno.org</a>)</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>: <cite style="font-style:italic">Das ist o.B.d.A. trivial!</cite> 2. Auflage. Vieweg Verlag, Wiesbaden 1992, ISBN 3-528-16442-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>13<span style="display:inline-block;width:.2em"> </span>f</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Hilfssatz&rft.au=Albrecht+Beutelspacher&rft.btitle=Das+ist+o.B.d.A.+trivial%21&rft.date=1992&rft.edition=2.&rft.genre=book&rft.isbn=3528164425&rft.pages=13f&rft.place=Wiesbaden&rft.pub=Vieweg+Verlag" style="display:none"> </span></span>

</li>

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