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<title>Élimination des quantificateurs</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Élimination des quantificateurs</h1>

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En <a href="Logique_math%C3%A9matique" title="Logique mathématique">logique mathématique</a>, ou plus précisément en <a href="Th%C3%A9orie_des_mod%C3%A8les" title="Théorie des modèles">théorie des modèles</a>, l'élimination des quantificateurs est l'action consistant à trouver une <a href="Formule_(math%C3%A9matiques)" title="Formule (mathématiques)">formule</a> sans quantificateur équivalente à une formule donnée contenant éventuellement des <a href="Quantificateur_(logique)" class="mw-redirect" title="Quantificateur (logique)">quantificateurs</a> dans la théorie considérée d'un certain langage.

Ainsi, si l'on considère la théorie des <a href="Corps_r%C3%A9el_clos" title="Corps réel clos">corps réels clos</a>, le langage de cette théorie est L={+,•,<,0,1} où +,• sont deux symboles de fonction d'arité 2, < est un symbole de <a href="Relation_binaire" title="Relation binaire">relation binaire</a>, et 0,1 sont deux symboles de constante, la L-formule ∃x (ax² + bx + c = 0) est équivalente à la L-formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a=0\land b=0\land c=0)\lor (a=0\land b\not =0)\lor (a\not =0\land \lnot (b^{2}-4ac<0))}">

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<annotation encoding="application/x-tex">{\displaystyle (a=0\land b=0\land c=0)\lor (a=0\land b\not =0)\lor (a\not =0\land \lnot (b^{2}-4ac<0))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656704311971e0e816d529b7583a1ef9d976cc15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.092ex; height:3.176ex;" alt="{\displaystyle (a=0\land b=0\land c=0)\lor (a=0\land b\not =0)\lor (a\not =0\land \lnot (b^{2}-4ac<0))}" loading="lazy"> dans la théorie, car dans cette théorie ax² + bx + c = 0 admet une solution si et seulement si a, b et c sont tous nuls, ou a est nul mais b est non nul, ou a non nul et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}-4ac}">

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<annotation encoding="application/x-tex">{\displaystyle b^{2}-4ac}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc2c88a48e0087a5786b460b2e856d118b5e23ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.291ex; height:2.843ex;" alt="{\displaystyle b^{2}-4ac}" loading="lazy"> est positif.

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

Soient L un langage et T une L-théorie, on dit que T admet l'élimination des quantificateurs si pour toute L-formule F, il existe une L-formule G sans quantificateur telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models F\leftrightarrow G}">

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<annotation encoding="application/x-tex">{\displaystyle T\models F\leftrightarrow G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f7c75bbcb534f0ccfa95bd991c139d4dae3c248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.123ex; height:2.843ex;" alt="{\displaystyle T\models F\leftrightarrow G}" loading="lazy">. Autrement dit, une théorie T du langage L admet l'élimination des quantificateurs si toute formule du langage L est équivalente à une formule sans quantificateur dans cette théorie.

<div class="mw-heading mw-heading2"><h2 id="Intérêt_d'élimination_des_quantificateurs">Intérêt d'élimination des quantificateurs</h2></div>

Les formules sans variables sont souvent plus faciles à décider ; l'algorithme d'élimination des quantificateurs peut donc servir de <a href="Probl%C3%A8me_de_d%C3%A9cision" title="Problème de décision">procédure de décision</a> pour cette théorie. Dans une théorie décidable admettant l'élimination des quantificateurs, il existe un algorithme qui en acceptant une formule donne toujours une formule sans quantificateur. Le seul problème est l'efficacité de l'algorithme.

Elle nous permet aussi de mieux comprendre les ensembles définissables dans une théorie.

<div class="mw-heading mw-heading2"><h2 id="Quelques_critères_d'élimination_des_quantificateurs">Quelques critères d'élimination des quantificateurs</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Critère_1">Critère 1</h3></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> une L-theorie.

Supposons que pour toute L-formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F({\bar {v}},w)}">

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<annotation encoding="application/x-tex">{\displaystyle F({\bar {v}},w)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e47a7ba16ad6a7660fa13dd460a0681ed2416db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.475ex; height:2.843ex;" alt="{\displaystyle F({\bar {v}},w)}" loading="lazy"> sans quantificateur il existe une L-formule sans quantificateur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G({\bar {v}})}">

<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3d39a6e33fe1dbfc17baed7c2c45bfa6eee6d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.863ex; height:2.843ex;" alt="{\displaystyle G({\bar {v}})}" loading="lazy"> telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models \exists wF({\bar {v}},w)\leftrightarrow G({\bar {v}})}">

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</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T\models \exists wF({\bar {v}},w)\leftrightarrow G({\bar {v}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5dd9f8ef0dd6b12ab834cfe06df78fd436f04fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.85ex; height:2.843ex;" alt="{\displaystyle T\models \exists wF({\bar {v}},w)\leftrightarrow G({\bar {v}})}" loading="lazy">.

Alors, T admet l'élimination des quantificateurs.

<div class="mw-heading mw-heading4"><h4 id="Preuve">Preuve</h4></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},x_{2},...,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle F(x_{1},x_{2},...,x_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b89476f01e192a1528aab873022fc6f05ab6892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.07ex; height:2.843ex;" alt="{\displaystyle F(x_{1},x_{2},...,x_{n})}" loading="lazy"> une L-formule. On montre par l'induction sur la complexité de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> qu'il existe une L-formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> sans quantificateur telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models F\leftrightarrow G}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\models F\leftrightarrow G}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f7c75bbcb534f0ccfa95bd991c139d4dae3c248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.123ex; height:2.843ex;" alt="{\displaystyle T\models F\leftrightarrow G}" loading="lazy">.

Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F|=0}">

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<annotation encoding="application/x-tex">{\displaystyle |F|=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46afe41873f69b951d85a9d04d0205fb8e9befb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.295ex; height:2.843ex;" alt="{\displaystyle |F|=0}" loading="lazy"> alors posons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=F}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25503f5597ff9ff634f8b4d07ce2e4cf01bb2402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.666ex; height:2.176ex;" alt="{\displaystyle G=F}" loading="lazy">. Supposons que la propriété est vraie pour toutes les formules de complexité strictement plus petite que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F|=n}">

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<annotation encoding="application/x-tex">{\displaystyle |F|=n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd911ff6f204da97c631dd18374ccbed490d90b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.528ex; height:2.843ex;" alt="{\displaystyle |F|=n}" loading="lazy">, il y a trois cas : soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:=\lnot H_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle F:=\lnot H_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59560f849377b1db6d04fe48246a74be075285bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.022ex; height:2.509ex;" alt="{\displaystyle F:=\lnot H_{1}}" loading="lazy">, alors par hypothèse d'induction il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"> sans quantificateur telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}" loading="lazy"> est équivalente à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}}">

<msub>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy">, il suffit de poser <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=\lnot H_{2}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=\lnot H_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17be9fcae82841af1e5517c57e7194ec61a69212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.108ex; height:2.509ex;" alt="{\displaystyle G:=\lnot H_{2}}" loading="lazy">; soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:=H_{1}\land H_{2}}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F:=H_{1}\land H_{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cd7cfda95fb4f99e47585e04a881a6a585c70e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.04ex; height:2.509ex;" alt="{\displaystyle F:=H_{1}\land H_{2}}" loading="lazy">, alors par hypothèse d'induction il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{3}}">

<msub>

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</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4bc9dc729e1d53cb3651e618e87861c26656f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{3}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{4}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{4}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf998dd6a6fe8dd490c0e468d118be09b8a11995.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{4}}" loading="lazy"> sans quantificateur telles que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models H_{1}\leftrightarrow H_{3}}">

<msub>

</mrow>

</msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T\models H_{1}\leftrightarrow H_{3}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba334a58ec3559b5ac9aacbe553d248869c12ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.526ex; height:2.843ex;" alt="{\displaystyle T\models H_{1}\leftrightarrow H_{3}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models H_{2}\leftrightarrow H_{4}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\models H_{2}\leftrightarrow H_{4}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2314287d1f82ea622d336e4b193647f908c0843f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.526ex; height:2.843ex;" alt="{\displaystyle T\models H_{2}\leftrightarrow H_{4}}" loading="lazy">, et on pose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=H_{3}\land H_{4}}">

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<annotation encoding="application/x-tex">{\displaystyle G:=H_{3}\land H_{4}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2159a20afef2cedd6c06a44f0db27080c44ac6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.126ex; height:2.509ex;" alt="{\displaystyle G:=H_{3}\land H_{4}}" loading="lazy">; soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:=\exists xH_{1}}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle F:=\exists xH_{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc82f5bb02ec3b18f91777d300c202ff9327689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.094ex; height:2.509ex;" alt="{\displaystyle F:=\exists xH_{1}}" loading="lazy">, par hypothèse de l'induction il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}" loading="lazy"> sans quantificateur telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models \exists xH_{1}\leftrightarrow \exists xH_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\models \exists xH_{1}\leftrightarrow \exists xH_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc31996346e813c5625ef66dc85d87ba6f84e302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.771ex; height:2.843ex;" alt="{\displaystyle T\models \exists xH_{1}\leftrightarrow \exists xH_{2}}" loading="lazy">, et par hypothèse, il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4bc9dc729e1d53cb3651e618e87861c26656f6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{3}}" loading="lazy"> sans quantificateur telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models \exists xH_{2}\leftrightarrow H_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\models \exists xH_{2}\leftrightarrow H_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bc70c20c537c659e08effd2fff18ccaf1724e52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.149ex; height:2.843ex;" alt="{\displaystyle T\models \exists xH_{2}\leftrightarrow H_{3}}" loading="lazy">, on pose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=H_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=H_{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b06f03c08fa6d29c86b256a63fc59079589e108e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.557ex; height:2.509ex;" alt="{\displaystyle G:=H_{3}}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Exemple:_DLO">Exemple: DLO</h4></div>

On montre que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a3563fb7d0e298239788eb4f0960d4354fc7c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.28ex; height:2.176ex;" alt="{\displaystyle DLO}" loading="lazy"> (Dense Linear Ordering) admet l’élimination des quantificateurs en vérifiant la condition du critère 1. Autrement dit, soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},\ldots ,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},\ldots ,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a39a69d44f4f8932a1388c8b6e9169ee39ad4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.85ex; height:2.843ex;" alt="{\displaystyle F(x_{1},\ldots ,x_{n},y)}" loading="lazy"> une formule sans quantificateur, cherchons une formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x_{1},\ldots ,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(x_{1},\ldots ,x_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3adbb73526200cff365e86e4cd79485410b584ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.746ex; height:2.843ex;" alt="{\displaystyle G(x_{1},\ldots ,x_{n})}" loading="lazy"> sans quantificateur telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models F\leftrightarrow G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models F\leftrightarrow G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff2e56c82213a4802733d2bd90d54a7aeacfd96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.767ex; height:2.843ex;" alt="{\displaystyle DLO\models F\leftrightarrow G}" loading="lazy">.

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> est sans quantificateur, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> est équivalente à une formule sous forme disjonctive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c150cd42c12d239d5caea2f69671d01f8fd155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.715ex; height:5.843ex;" alt="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i,j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f379f42961dd620c7a05dc1c538117ec105877d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.678ex; height:2.843ex;" alt="{\displaystyle A_{i,j}}" loading="lazy"> sont des formules atomiques (ou leur négation) du langage de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a3563fb7d0e298239788eb4f0960d4354fc7c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.28ex; height:2.176ex;" alt="{\displaystyle DLO}" loading="lazy">. Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20cd4fb4e3337c0f0945aede355ac4fb738730d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.301ex; height:5.843ex;" alt="{\displaystyle DLO\models \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models \bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models \bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65914393f366ebf8452296b8d84c85bd0ebaa4e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.332ex; height:5.843ex;" alt="{\displaystyle DLO\models \bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}" loading="lazy"> pour un certain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> est équivalente à une formule de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e4dc9488e69a6fcbb403946d9bb09846d42bffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.612ex; height:5.509ex;" alt="{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> sont des formules atomiques (ou leur négation) du langage de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a3563fb7d0e298239788eb4f0960d4354fc7c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.28ex; height:2.176ex;" alt="{\displaystyle DLO}" loading="lazy">.

Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models \lnot (x<y)\leftrightarrow (y<x\lor x=y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models \lnot (x<y)\leftrightarrow (y<x\lor x=y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae25e006782523a6d86b5f149d49db5f3dbe5939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.702ex; height:2.843ex;" alt="{\displaystyle DLO\models \lnot (x<y)\leftrightarrow (y<x\lor x=y)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models \lnot (x=y)\leftrightarrow (x<y\lor y<x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models \lnot (x=y)\leftrightarrow (x<y\lor y<x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37c7b4d5ce73a51f121ab550723bb9c82a5b49c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.702ex; height:2.843ex;" alt="{\displaystyle DLO\models \lnot (x=y)\leftrightarrow (x<y\lor y<x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models x=x\leftrightarrow \top }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models x=x\leftrightarrow \top }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ad94ac41a9d5679a087d9a7a11d1341d40190cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.766ex; height:2.843ex;" alt="{\displaystyle DLO\models x=x\leftrightarrow \top }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models x<x\leftrightarrow \bot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models x<x\leftrightarrow \bot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89b660a7fa2971b6cab50ea02600a5a231265cbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.766ex; height:2.843ex;" alt="{\displaystyle DLO\models x<x\leftrightarrow \bot }" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models H\land \top \leftrightarrow H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models H\land \top \leftrightarrow H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc4e132ba1e6487e2045a147be1f202543af06e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.718ex; height:2.843ex;" alt="{\displaystyle DLO\models H\land \top \leftrightarrow H}" loading="lazy">, on peut supposer que les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> sont de la forme : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x_{i},x_{i}=x_{j},y<x_{i},x_{i}<y,x_{i}<x_{j}}">

<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 y=x_{i},x_{i}=x_{j},y<x_{i},x_{i}<y,x_{i}<x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1743ee7069e7f6cfdcb6b22606d0ba2afbd565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.22ex; height:2.509ex;" alt="{\displaystyle y=x_{i},x_{i}=x_{j},y<x_{i},x_{i}<y,x_{i}<x_{j}}" loading="lazy">, ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot .}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dcd49f9cc2b5d71f00a724f650f0f295b908e0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \bot .}" loading="lazy">

S'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}=\bot }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}=\bot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b81aea26a1a462b6691c6d1e067c617054d6c8e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.449ex; height:2.509ex;" alt="{\displaystyle A_{i}=\bot }" loading="lazy"> alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)\leftrightarrow \bot }">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)\leftrightarrow \bot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/541188ac4cb5e31f9c4837526931682f05a16342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.62ex; height:5.509ex;" alt="{\displaystyle DLO\models \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)\leftrightarrow \bot }" loading="lazy">, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=\bot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=\bot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe285c4bf2b4c03633d1d750cc5c2b64ac172a85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.38ex; height:2.176ex;" alt="{\displaystyle G:=\bot }" loading="lazy"> convient. Supposons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}\not =\bot }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}\not =\bot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ffa1fb055d27c6d8eb7b25e80214d07eef74a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.449ex; height:2.676ex;" alt="{\displaystyle A_{i}\not =\bot }" loading="lazy"> pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">.

S'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/480208be374db35ca5430afab6813a076ffb41ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.383ex; height:2.009ex;" alt="{\displaystyle y=x_{i}}" loading="lazy"> alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=F[x_{i}/y]}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=F[x_{i}/y]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5c6ebc27144f147a90b0a0a39a9ff96c6b2f31c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.054ex; height:2.843ex;" alt="{\displaystyle G:=F[x_{i}/y]}" loading="lazy"> convient. Donc on peut supposer que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> n'est pas de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/480208be374db35ca5430afab6813a076ffb41ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.383ex; height:2.009ex;" alt="{\displaystyle y=x_{i}}" loading="lazy"> pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">.

Donc les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> sont de la forme : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=x_{j},y<x_{i},x_{i}<y}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{j},y<x_{i},x_{i}<y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e17d869cbf9742947c11a3fb09375e7ad6603c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.302ex; height:2.509ex;" alt="{\displaystyle x_{i}=x_{j},y<x_{i},x_{i}<y}" loading="lazy">, ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}<x_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}<x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c74cef9d1c739958217238ad7d03ab915c59f03d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.509ex;" alt="{\displaystyle x_{i}<x_{j}}" loading="lazy">.

Ainsi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y):=\bigwedge _{i\in I}A_{i}\land \bigwedge _{j\in J}A_{j}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y):=\bigwedge _{i\in I}A_{i}\land \bigwedge _{j\in J}A_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0649fd738ab4b15d245405baee6b836be73d574e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.305ex; height:5.843ex;" alt="{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y):=\bigwedge _{i\in I}A_{i}\land \bigwedge _{j\in J}A_{j}}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> sont de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=x_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4e90e70df81dd461038b628870de3cf7dfa0cc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.343ex;" alt="{\displaystyle x_{i}=x_{j}}" loading="lazy"> ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}<x_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}<x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c74cef9d1c739958217238ad7d03ab915c59f03d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.509ex;" alt="{\displaystyle x_{i}<x_{j}}" loading="lazy">, et les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6019bb70c912e59e9d5f442e9217517743ed4831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}" loading="lazy"> sont de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2cbdbf2ae8e01416a03c105083ddccfd589a62c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.383ex; height:2.176ex;" alt="{\displaystyle y<x_{i}}" loading="lazy"> ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}<y}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e618b60675d96aa41dba286aaece77ebfdedce42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.383ex; height:2.176ex;" alt="{\displaystyle x_{i}<y}" loading="lazy">.

On a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j\in J}A_{j}:=\bigwedge _{j\in S}x_{j}<y\land \bigwedge _{j\in T}y<x_{j}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j\in J}A_{j}:=\bigwedge _{j\in S}x_{j}<y\land \bigwedge _{j\in T}y<x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d26119a79175c7ca708c7361d6dfaae404ceea6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:31.726ex; height:5.843ex;" alt="{\displaystyle \bigwedge _{j\in J}A_{j}:=\bigwedge _{j\in S}x_{j}<y\land \bigwedge _{j\in T}y<x_{j}}" loading="lazy">. Donc on a deux cas :

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\cap T\not =\emptyset }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d86b77e857db65571572a5189f0488c1416daebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.979ex; height:2.843ex;" alt="{\displaystyle S\cap T\not =\emptyset }" loading="lazy">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=\bot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=\bot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe285c4bf2b4c03633d1d750cc5c2b64ac172a85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.38ex; height:2.176ex;" alt="{\displaystyle G:=\bot }" loading="lazy"> convient.</li></ul>

<ul><li>Sinon si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\emptyset }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a7c38be46a9b3fe15e5893edc24def5ba893e5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.76ex; height:2.509ex;" alt="{\displaystyle S=\emptyset }" loading="lazy"> ou <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\emptyset }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37e44fc957dca841d6628dc3008ead4a7bb95db7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.897ex; height:2.509ex;" alt="{\displaystyle T=\emptyset }" loading="lazy">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=\top }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=\top }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb557852834d3e82495248cde0c9fb39a7b4078a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.38ex; height:2.176ex;" alt="{\displaystyle G:=\top }" loading="lazy"> convient car l'ordre est sans extrémités.</li></ul>

<ul><li>Sinon <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DLO\models \bigwedge _{j\in J}A_{j}\leftrightarrow \bigwedge _{i\in S,j\in T}x_{i}<x_{j}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DLO\models \bigwedge _{j\in J}A_{j}\leftrightarrow \bigwedge _{i\in S,j\in T}x_{i}<x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e53690b758a07341011dd4f87d8d73454b8cfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.02ex; height:5.843ex;" alt="{\displaystyle DLO\models \bigwedge _{j\in J}A_{j}\leftrightarrow \bigwedge _{i\in S,j\in T}x_{i}<x_{j}}" loading="lazy"> car l'ordre est dense, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G:=\bigwedge _{i\in I}A_{i}\land \bigwedge _{i\in S,j\in T}x_{i}<x_{j}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G:=\bigwedge _{i\in I}A_{i}\land \bigwedge _{i\in S,j\in T}x_{i}<x_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb67a8c570d50d82c4718d9ffe7dc9082fd7c368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.632ex; height:5.843ex;" alt="{\displaystyle G:=\bigwedge _{i\in I}A_{i}\land \bigwedge _{i\in S,j\in T}x_{i}<x_{j}}" loading="lazy"> convient.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Critère_2">Critère 2</h3></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> une L−théorie.

Supposons que pour toute L-formule sans quantificateur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},x_{2},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},x_{2},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/350923289f26b858365e0d26486804c5a213fc0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.259ex; height:2.843ex;" alt="{\displaystyle F(x_{1},x_{2},...,x_{n},y)}" loading="lazy">, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> sont deux modèles de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> est une sous-structure commune de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},...,a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},...,a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56d574ff1f6ac680957241f7b55f7bf72ecde023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.22ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},...,a_{n}}" loading="lazy"> des éléments de l’ensemble de base de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, et il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58256b69a6718e8dedba9f3d851c0fe33c166671.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.28ex; height:2.176ex;" alt="{\displaystyle b\in M}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},a_{2},...a_{n},b)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},a_{2},...a_{n},b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6247e14d6e671a971edfde40624278dbe62789a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.515ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},a_{2},...a_{n},b)}" loading="lazy">, alors il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> dans le domaine de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models F(a_{1},a_{2},...,a_{n},c)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models F(a_{1},a_{2},...,a_{n},c)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a806035a05705a291ddcd84ecfbb36a6df894c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.18ex; height:2.843ex;" alt="{\displaystyle N\models F(a_{1},a_{2},...,a_{n},c)}" loading="lazy">.

Alors, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> admet l’élimination des quantificateurs.

<div class="mw-heading mw-heading4"><h4 id="Preuve_de_la_réciproque">Preuve de la réciproque</h4></div>

Montrons la réciproque (la preuve directe est plus délicate). Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f459119c1c9c030f07c7f4d7f24c5ed675e3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.841ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},y)}" loading="lazy"> une L-formule sans quantificateur. Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> deux modèles de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> une sous-structure commune de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},...a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},...a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f07b506322ad173f1877e71b5362a9f769ae66b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.186ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},...a_{n}}" loading="lazy"> des éléments de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. Supposons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models \exists yF(a_{1},a_{2},...a_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models \exists yF(a_{1},a_{2},...a_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13da82e937829d4d1b7666b8cfd30d477aee993b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.121ex; height:2.843ex;" alt="{\displaystyle M\models \exists yF(a_{1},a_{2},...a_{n},y)}" loading="lazy">. Alors par élimination des quantificateurs, il existe une L-formule sans quantificateur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x_{1},...,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(x_{1},...,x_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c8bd671cb55b7b8103095d128e3663ef3a983c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.738ex; height:2.843ex;" alt="{\displaystyle G(x_{1},...,x_{n})}" loading="lazy"> telle que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\models \exists yF(x_{1},...,x_{n},y)\leftrightarrow G(x_{1},...,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\models \exists yF(x_{1},...,x_{n},y)\leftrightarrow G(x_{1},...,x_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/010683356af7810172386ef96b81bf90538d9bb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.583ex; height:2.843ex;" alt="{\displaystyle T\models \exists yF(x_{1},...,x_{n},y)\leftrightarrow G(x_{1},...,x_{n})}" loading="lazy">, et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models G(a_{1},...,a_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models G(a_{1},...,a_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eb713f87663714548b18314bec41e1f7f5c7b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.285ex; height:2.843ex;" alt="{\displaystyle M\models G(a_{1},...,a_{n})}" loading="lazy">. Or, comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> est une sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x_{1},...,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(x_{1},...,x_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c8bd671cb55b7b8103095d128e3663ef3a983c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.738ex; height:2.843ex;" alt="{\displaystyle G(x_{1},...,x_{n})}" loading="lazy"> est sans quantificateur, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models G(a_{1},...,a_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models G(a_{1},...,a_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eb713f87663714548b18314bec41e1f7f5c7b0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.285ex; height:2.843ex;" alt="{\displaystyle M\models G(a_{1},...,a_{n})}" loading="lazy"> équivaut à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\models G(a_{1},...,a_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\models G(a_{1},...,a_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52ef5dd397d3e4906ac41394d622b3fb1f737ad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.586ex; height:2.843ex;" alt="{\displaystyle A\models G(a_{1},...,a_{n})}" loading="lazy">, équivalant, de même, à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models G(a_{1},...,a_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models G(a_{1},...,a_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eacf550a7f492931a4ff9d055090ad376429110e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.907ex; height:2.843ex;" alt="{\displaystyle N\models G(a_{1},...,a_{n})}" loading="lazy">, d'où enfin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models \exists yF(a_{1},a_{2},...a_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models \exists yF(a_{1},a_{2},...a_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/085ea3d783d6466a5b29c0411242eec2b38496d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.742ex; height:2.843ex;" alt="{\displaystyle N\models \exists yF(a_{1},a_{2},...a_{n},y)}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Exemple:_DAG">Exemple: DAG</h4></div>

On montre que la théorie de <a href="Groupe_ab%C3%A9lien" title="Groupe abélien">groupe abélien</a> divisible sans torsion (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DAG}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b1c8e18be7c57ddfaab188c6fba11c327145cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.494ex; height:2.176ex;" alt="{\displaystyle DAG}" loading="lazy">) admet l’élimination des quantificateurs en montrant qu'elle vérifie la condition du critère 2.

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f459119c1c9c030f07c7f4d7f24c5ed675e3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.841ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},y)}" loading="lazy"> une formule sans quantificateur. Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> deux groupes abéliens divisibles sans torsion, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> un sous-groupe commun de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},...,a_{n})\in G^{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{1},...,a_{n})\in G^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7b9a748e2c0e8429dc4049a24ab26f2ecd4f50c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.597ex; height:2.843ex;" alt="{\displaystyle (a_{1},...,a_{n})\in G^{n}}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4c521df89304273881290faf7338f7017af1bca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.622ex; height:2.176ex;" alt="{\displaystyle h\in M}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},...,a_{n},h)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},...,a_{n},h)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a68ac9f467654850d25bbe8f05c3758a4ac9cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.572ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},...,a_{n},h)}" loading="lazy">. On montre d'abord qu'il existe un sous-groupe commun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> est un sous-groupe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">, puis on montre que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\models \exists xF(a_{1},...,a_{n},x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\models \exists xF(a_{1},...,a_{n},x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5960572a1324028756389bd0d9b3fb23ef7c811c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.807ex; height:2.843ex;" alt="{\displaystyle H\models \exists xF(a_{1},...,a_{n},x)}" loading="lazy"> avant de conclure.

Montrons qu'il existe un sous-groupe commun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> est un sous-groupe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">.

Posons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=G\times N^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:=G\times N^{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06652c4bfdf274b2abab8aa018ba356f6ac2f8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.57ex; height:2.343ex;" alt="{\displaystyle X:=G\times N^{*}}" loading="lazy">. On définit une relation d'équivalence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy"> sur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> par : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g,n)\sim (h,m)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g,n)\sim (h,m)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ba4bd597acce2c21d50caea011e51a847abf502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.675ex; height:2.843ex;" alt="{\displaystyle (g,n)\sim (h,m)}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mg=nh}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9db1b85541fbf8801ad71fab42c49a879618b240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.989ex; height:2.509ex;" alt="{\displaystyle mg=nh}" loading="lazy">. Posons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=X/\sim }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=X/\sim }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18589fe98603e99f3982692272509bcaca72339e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.758ex; height:2.843ex;" alt="{\displaystyle H=X/\sim }" loading="lazy">. Notons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,n))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d3472b7faee05f70f8f58cf28c107cb100e3058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.163ex; height:2.843ex;" alt="{\displaystyle ((g,n))}" loading="lazy"> la classe d'équivalence de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g,n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g,n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86e53d0c7140c3165a39b6352c9ae92f0bef653f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.354ex; height:2.843ex;" alt="{\displaystyle (g,n)}" loading="lazy">. On définit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +:H^{2}\rightarrow H}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dacdd417c9e342246584b7bce31c87e1cd4d683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.581ex; height:2.843ex;" alt="{\displaystyle +:H^{2}\rightarrow H}" loading="lazy"> en posant, pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,n),(h,m))\in H^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,n),(h,m))\in H^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0a00fa2a523b2e27b62b38ab4c746681e8a8ca0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.418ex; height:3.176ex;" alt="{\displaystyle ((g,n),(h,m))\in H^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,n))+((h,m))=((mg+nh,mn))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,n))+((h,m))=((mg+nh,mn))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b44d0dd4f55b9305f4b6eecd1940317d8af4c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.952ex; height:2.843ex;" alt="{\displaystyle ((g,n))+((h,m))=((mg+nh,mn))}" loading="lazy">.

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"> est bien définie : Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g',n')\in ((g,n))}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g',n')\in ((g,n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/986d6f87fceb154e9cc0b7b492bd42652cd17aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.73ex; height:3.009ex;" alt="{\displaystyle (g',n')\in ((g,n))}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h',m')\in ((h,m))}">

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (h',m')\in ((h,m))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dfcba4b8c6b6658cf4e7da16f00571a254a72e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.465ex; height:3.009ex;" alt="{\displaystyle (h',m')\in ((h,m))}" loading="lazy">. On a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g',n'))+((h',m'))=((m'g'+n'h',m'n')).}">

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g',n'))+((h',m'))=((m'g'+n'h',m'n')).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/282508d3a573583f77b417e1f3a0ede1133bff6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.451ex; height:3.009ex;" alt="{\displaystyle ((g',n'))+((h',m'))=((m'g'+n'h',m'n')).}" loading="lazy"> Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m'n'(mg+nh)=m'n'mg+m'n'nh=m'mn'g+n'nm'h=mnm'g'+mnn'h=mn(mg'+n'h)}">

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m'n'(mg+nh)=m'n'mg+m'n'nh=m'mn'g+n'nm'h=mnm'g'+mnn'h=mn(mg'+n'h)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7f035855704f73d593783255b33caff5cf855cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:97.421ex; height:3.009ex;" alt="{\displaystyle m'n'(mg+nh)=m'n'mg+m'n'nh=m'mn'g+n'nm'h=mnm'g'+mnn'h=mn(mg'+n'h)}" loading="lazy">, on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((m'g'+n'h',m'n'))=((mg+nh,mn))}">

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((m'g'+n'h',m'n'))=((mg+nh,mn))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f4b6ce1ca4c532731e104ab0700da2c95e91252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.845ex; height:3.009ex;" alt="{\displaystyle ((m'g'+n'h',m'n'))=((mg+nh,mn))}" loading="lazy">.</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (H,+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1b5e04d98957dae89ad5e4aed4c9c1f4e17a7af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.715ex; height:2.843ex;" alt="{\displaystyle (H,+)}" loading="lazy"> est un groupe : Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((g,n)),((h,m)),((k,l)))\in H^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (((g,n)),((h,m)),((k,l)))\in H^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7ed772c9b04dd6dc78bdd3b13ce35496f88b6c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.628ex; height:3.176ex;" alt="{\displaystyle (((g,n)),((h,m)),((k,l)))\in H^{3}}" loading="lazy">. On a :</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((0,1))+((g,n))=((g,n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaa5c22bb75e406c24ebadf1b614c0282c5c8e3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.243ex; height:2.843ex;" alt="{\displaystyle ((0,1))+((g,n))=((g,n))}" loading="lazy">;

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((-g,n))+((g,n))=((0,nn))=((0,1))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05e3820318b3cbce48fc273c40490230e3645c72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.754ex; height:2.843ex;" alt="{\displaystyle ((-g,n))+((g,n))=((0,nn))=((0,1))}" loading="lazy">;

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (((g,n))+((h,m)))+((k,l))=((g,n))+(((h,m))+((k,l)))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9886ef2092902aa148642259aaa75ae6243e2229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.583ex; height:2.843ex;" alt="{\displaystyle (((g,n))+((h,m)))+((k,l))=((g,n))+(((h,m))+((k,l)))}" loading="lazy">.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (H,+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1b5e04d98957dae89ad5e4aed4c9c1f4e17a7af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.715ex; height:2.843ex;" alt="{\displaystyle (H,+)}" loading="lazy"> est sans torsion : Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,n))\in H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,n))\in H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2848aaddce34673324dd1cad7498feac2a1bb6d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.068ex; height:2.843ex;" alt="{\displaystyle ((g,n))\in H}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {N} ^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/081e0d86d96728ac8dd64b6eebdde7e05019edeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.613ex; height:2.343ex;" alt="{\displaystyle m\in \mathbb {N} ^{*}}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m((g,n))=((0,1))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m((g,n))=((0,1))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74f65671dc9bf5ddf4458d6f77db68c4758f361d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.28ex; height:2.843ex;" alt="{\displaystyle m((g,n))=((0,1))}" loading="lazy">. On a donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((mg,n))=((0,k)).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((mg,n))=((0,k)).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3d87b8f2531b37c4a0869613b82f440e54d9a23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.975ex; height:2.843ex;" alt="{\displaystyle ((mg,n))=((0,k)).}" loading="lazy"> Donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kmg=(km)g=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle kmg=(km)g=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d0860fa869edae0eaecb7080c9e8b66a88297ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.904ex; height:2.843ex;" alt="{\displaystyle kmg=(km)g=0}" loading="lazy">. Ainsi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0607b4f563220901d455691e4a1705d598fdbaeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g=0}" loading="lazy"> car <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> est sans torsion.</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (H,+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1b5e04d98957dae89ad5e4aed4c9c1f4e17a7af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.715ex; height:2.843ex;" alt="{\displaystyle (H,+)}" loading="lazy"> est divisible : Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,n))\in H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,n))\in H}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2848aaddce34673324dd1cad7498feac2a1bb6d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.068ex; height:2.843ex;" alt="{\displaystyle ((g,n))\in H}" loading="lazy">. On a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m((g,mn))=((g,n))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m((g,mn))=((g,n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/350e72f266550a685605df6286f2221a457c695e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.506ex; height:2.843ex;" alt="{\displaystyle m((g,mn))=((g,n))}" loading="lazy">.</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (H,+)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1b5e04d98957dae89ad5e4aed4c9c1f4e17a7af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.715ex; height:2.843ex;" alt="{\displaystyle (H,+)}" loading="lazy"> est abélien : Soit<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((g,m)),((h,n)))\in H^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (((g,m)),((h,n)))\in H^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0cc447fd127b492738600de2864e0c804493c4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.037ex; height:3.176ex;" alt="{\displaystyle (((g,m)),((h,n)))\in H^{2}}" loading="lazy">. On a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,m))+((h,n))=((ng+mh,mn))=((mh+ng,mn))=((h,n))+((g,m))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,m))+((h,n))=((ng+mh,mn))=((mh+ng,mn))=((h,n))+((g,m))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/faf05fec1f98692e3f0f96814f7ef8dc737f8b25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.003ex; height:2.843ex;" alt="{\displaystyle ((g,m))+((h,n))=((ng+mh,mn))=((mh+ng,mn))=((h,n))+((g,m))}" loading="lazy">.</li></ul>

Montrons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:G\rightarrow H,\ g\mapsto ((g,1))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:G\rightarrow H,\ g\mapsto ((g,1))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b892d4acd6d83ebb19bea81d89f52a6547a0dc19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.996ex; height:2.843ex;" alt="{\displaystyle f:G\rightarrow H,\ g\mapsto ((g,1))}" loading="lazy"> est un homomorphisme injectif. On a :

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=((0,1))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/156cffefda5678f8624021e41e115121fe6e7503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.326ex; height:2.843ex;" alt="{\displaystyle f(0)=((0,1))}" loading="lazy">;

Pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g,h)\in G^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g,h)\in G^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a17ca514a4dd4ba0092ad4fcccb222bbfd29f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.02ex; height:3.176ex;" alt="{\displaystyle (g,h)\in G^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g+h)=((g+h,1))=((g,1))+((h,1))=f(g)+f(h)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(g+h)=((g+h,1))=((g,1))+((h,1))=f(g)+f(h)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2a10e12968b80d9778e6fae5b7938f7e46d2695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.185ex; height:2.843ex;" alt="{\displaystyle f(g+h)=((g+h,1))=((g,1))+((h,1))=f(g)+f(h)}" loading="lazy">;

Pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g,h)\in G^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g,h)\in G^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a17ca514a4dd4ba0092ad4fcccb222bbfd29f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.02ex; height:3.176ex;" alt="{\displaystyle (g,h)\in G^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=h}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50f4fd70b1200ffb96f9b7dccc5b64296e8da9f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.553ex; height:2.509ex;" alt="{\displaystyle g=h}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((x,1))=((y,1)).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((x,1))=((y,1)).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ed4bd712b1648b3b2b21cc0bc35fa96944e893a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.86ex; height:2.843ex;" alt="{\displaystyle ((x,1))=((y,1)).}" loading="lazy">

Montrons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:H\rightarrow M,\ ((g,n))\mapsto g/n}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:H\rightarrow M,\ ((g,n))\mapsto g/n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bedc922b29634f41b171feb9b45aa6bb786aac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.461ex; height:2.843ex;" alt="{\displaystyle h:H\rightarrow M,\ ((g,n))\mapsto g/n}" loading="lazy"> est bien défini et est un homomorphisme injectif.

</mstyle>

</mrow>

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

</semantics>

</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"> est bien défini : Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k,m)\in ((g,n))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k,m)\in ((g,n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/782ac87de43b1aa43388bb8b226220db21cc4790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.099ex; height:2.843ex;" alt="{\displaystyle (k,m)\in ((g,n))}" loading="lazy">. Alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mg=nk}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21648a53ed14486ff39e5113be21b167b4d008a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle mg=nk}" loading="lazy">. Donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g,n)=g/n=(mg)/mn=(nk/mn)=k/m.}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(g,n)=g/n=(mg)/mn=(nk/mn)=k/m.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d868a69e663fe0661c366390b3aac3c3f9df3e86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.337ex; height:2.843ex;" alt="{\displaystyle f(g,n)=g/n=(mg)/mn=(nk/mn)=k/m.}" loading="lazy">

</mstyle>

</mrow>

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

</semantics>

</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"> est un homomorphisme injectif : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(((0,1)))=0/1=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(((0,1)))=0/1=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e5cfb857f16ba30c94ed081b2d4172b32f9a02d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.972ex; height:2.843ex;" alt="{\displaystyle h(((0,1)))=0/1=0}" loading="lazy">;

Pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((g,m)),((k,n)))\in H^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (((g,m)),((k,n)))\in H^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e847eab4f5a2fbab29de39298d265768cf6e1e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.909ex; height:3.176ex;" alt="{\displaystyle (((g,m)),((k,n)))\in H^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(((g,m))+((k,n)))=h(((ng+mk,mn)))=(ng+mk)/(mn)=g/m+k/n=h(((g,m)))+h(((k,n)));}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(((g,m))+((k,n)))=h(((ng+mk,mn)))=(ng+mk)/(mn)=g/m+k/n=h(((g,m)))+h(((k,n)));}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b64a767a029da5d2675236e23e9b5e68dcd4c12f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:105.886ex; height:2.843ex;" alt="{\displaystyle h(((g,m))+((k,n)))=h(((ng+mk,mn)))=(ng+mk)/(mn)=g/m+k/n=h(((g,m)))+h(((k,n)));}" loading="lazy">

Pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((g,m)),((k,n)))\in H^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (((g,m)),((k,n)))\in H^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e847eab4f5a2fbab29de39298d265768cf6e1e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.909ex; height:3.176ex;" alt="{\displaystyle (((g,m)),((k,n)))\in H^{2}}" loading="lazy">, si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((g,m))=((k,n))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((g,m))=((k,n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3f04baae4dfcc3240caf927c6bb04f91fec18d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.166ex; height:2.843ex;" alt="{\displaystyle ((g,m))=((k,n))}" loading="lazy">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ng=mk}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8bfb0db6772e3cb084271f7875033c31b0b8acf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.861ex; height:2.509ex;" alt="{\displaystyle ng=mk}" loading="lazy">, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mn(g/m)=mn(k/n)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle mn(g/m)=mn(k/n)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0a7b98e4dbb2c01f7af727c4c0b68a5ee037b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.675ex; height:2.843ex;" alt="{\displaystyle mn(g/m)=mn(k/n)}" loading="lazy">, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g/m=k/n}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g/m=k/n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a146092807e9a5a636b5bc59026d8d04b470056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.186ex; height:2.843ex;" alt="{\displaystyle g/m=k/n}" loading="lazy">; si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g/m=k/n}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g/m=k/n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a146092807e9a5a636b5bc59026d8d04b470056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.186ex; height:2.843ex;" alt="{\displaystyle g/m=k/n}" loading="lazy">, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ng=nm(g/m))=nm(k/n)=mk.}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ng=nm(g/m))=nm(k/n)=mk.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df0bf7f0c9c38abc78cfb0087b30a5746db9f9a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.185ex; height:2.843ex;" alt="{\displaystyle ng=nm(g/m))=nm(k/n)=mk.}" loading="lazy">

De même, il existe aussi un homomorphisme injectif de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. Ainsi on peut identifier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> avec un sous-groupe commun à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> contenant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

Montrons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\models \exists xF(a_{1},a_{2},...,a_{n},x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\models \exists xF(a_{1},a_{2},...,a_{n},x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6ea787b81a4336d538d60843a48a1d02bf2c346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.125ex; height:2.843ex;" alt="{\displaystyle H\models \exists xF(a_{1},a_{2},...,a_{n},x)}" loading="lazy">.

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f459119c1c9c030f07c7f4d7f24c5ed675e3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.841ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},y)}" loading="lazy"> est une formule sans quantificateur, donc elle est équivalente à une formule sous forme disjonctive : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c150cd42c12d239d5caea2f69671d01f8fd155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.715ex; height:5.843ex;" alt="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i,j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f379f42961dd620c7a05dc1c538117ec105877d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.678ex; height:2.843ex;" alt="{\displaystyle A_{i,j}}" loading="lazy"> sont des formules atomiques ou des négations de formules atomiques du langage.

Quand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0c150cd42c12d239d5caea2f69671d01f8fd155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.715ex; height:5.843ex;" alt="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{i,j}(x_{1},...,x_{n},y)}" loading="lazy"> est satisfait, il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91677c698b4d3f062d76d9e43ad3e914d243e758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.857ex; height:2.509ex;" alt="{\displaystyle i_{0}}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x{n},y)}">

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0629f1ff8673a7a0bf1a4ec88cb1a9376b077bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.755ex; height:5.843ex;" alt="{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x{n},y)}" loading="lazy"> est satisfait.

Comme dans le langage le seul symbole de prédicat est le symbole <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/505a4ceef454c69dffd23792c84b90f488543743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}" loading="lazy"> et le seul symbole de fonction est <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy">, une <a href="Formule_atomique" title="Formule atomique">formule atomique</a> dans ce langage est de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}(m_{i,j}x_{i}+m_{j}y=0)}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}(m_{i,j}x_{i}+m_{j}y=0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f168165eb76e8be536bca44d767999767679633.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.475ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}(m_{i,j}x_{i}+m_{j}y=0)}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i,j},m_{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i,j},m_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17bf038ce70220aa13c2da19087113d3ba269120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.959ex; height:2.343ex;" alt="{\displaystyle m_{i,j},m_{j}}" loading="lazy"> sont dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy">. Ainsi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x_{n},y):=(\bigwedge _{j\in S}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y=0)\land \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y\not =0))}">

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x_{n},y):=(\bigwedge _{j\in S}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y=0)\land \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y\not =0))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b809d63c41517f145cc69726113563c04954e53d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:81.205ex; height:7.176ex;" alt="{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x_{n},y):=(\bigwedge _{j\in S}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y=0)\land \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y\not =0))}" loading="lazy">.

S'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5979738445478e8f97b1419d7098a677450e5b96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.325ex; height:2.509ex;" alt="{\displaystyle j\in S}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{j}\not =0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{j}\not =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/409345e496800d9b059da68b0bd570926d0a8893.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.211ex; height:2.843ex;" alt="{\displaystyle m_{j}\not =0}" loading="lazy">, alors comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},...,a_{n},b)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},...,a_{n},b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38ea60694a91be0bfdfd20d652a14ffd1345967d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.231ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},...,a_{n},b)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\frac {\sum _{i=1}^{n}m_{i,j}(-a_{i})}{m_{j}}}\in G}">

<mfrac>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b={\frac {\sum _{i=1}^{n}m_{i,j}(-a_{i})}{m_{j}}}\in G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/517e283dc43bfddd96c267c9df9288d9f2be352e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.962ex; height:6.509ex;" alt="{\displaystyle b={\frac {\sum _{i=1}^{n}m_{i,j}(-a_{i})}{m_{j}}}\in G}" loading="lazy"> car G est un <a href="Groupe_divisible" title="Groupe divisible">groupe divisible</a>. Donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1c6817f334374a49bff9aeb4f8b94e7df66cc18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.902ex; height:2.176ex;" alt="{\displaystyle b\in H}" loading="lazy">.

Sinon, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x_{n},y):=\bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y\not =0).}">

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x_{n},y):=\bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y\not =0).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9261ad3a3f18ef4cdd2bb0092ae29fb6fd20fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.764ex; height:7.176ex;" alt="{\displaystyle \bigwedge _{j}A_{i_{0},j}(x_{1},...,x_{n},y):=\bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}x_{i}+m_{j}y\not =0).}" loading="lazy"> Comme H est sans torsion, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> est infini, car sinon pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in H,\ |H|x=0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in H,\ |H|x=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24361a060fae4e59dabd64284da1e1e77cca65b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.796ex; height:2.843ex;" alt="{\displaystyle x\in H,\ |H|x=0}" loading="lazy">. Comme pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\in T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2be35312e4f572c32c33370c840c4fffd8ca58a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.462ex; height:2.509ex;" alt="{\displaystyle j\in T}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{w\in H:\ \sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y=0\}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{w\in H:\ \sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y=0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff1ba739812703c1df47ee56b50d44826c19fc8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.752ex; height:6.843ex;" alt="{\displaystyle \{w\in H:\ \sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y=0\}}" loading="lazy"> est fini, il existe un élément dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> qui satisfait <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y\not =0).}">

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y\not =0).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8af1c615136b4f2c3c634ee2e8875a726d9523a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:26.359ex; height:7.176ex;" alt="{\displaystyle \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y\not =0).}" loading="lazy">

Comment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> est un sous groupe de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, il existe un élément de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> qui satisfait <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y\not =0)}">

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y\not =0)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a18987f94d34a361ae2a6dddd5613b0c82d5be67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:25.712ex; height:7.176ex;" alt="{\displaystyle \bigwedge _{j\in T}(\sum _{i=1}^{n}m_{i,j}a_{i}+m_{j}y\not =0)}" loading="lazy">. Par conséquent, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models \exists F(a_{1},...,a_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models \exists F(a_{1},...,a_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bce26bc35feb7dca4a4f607b786420e118a9b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.303ex; height:2.843ex;" alt="{\displaystyle N\models \exists F(a_{1},...,a_{n},y)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Critère_3">Critère 3</h3></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> une L−théorie telle que

1. Pour toute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\models T_{\forall }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\models T_{\forall }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab619c438c031a40c47eee36ac5f20195f7f19d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.552ex; height:2.843ex;" alt="{\displaystyle A\models T_{\forall }}" loading="lazy">, il existe une <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\models T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\models T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f8a59928a27817a8dc8d455d6bb87fe54ab6ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle K\models T}" loading="lazy"> et un monomorphisme<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:A\rightarrow K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i:A\rightarrow K}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a43549fc90ad773a248c7eb5be4ab3d5c0a5345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.163ex; height:2.176ex;" alt="{\displaystyle i:A\rightarrow K}" loading="lazy"> tels que pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9dccbe62f90dbd2298fb8f28d4db21b735a76408.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.005ex; height:2.843ex;" alt="{\displaystyle N\models T}" loading="lazy"> et monomorphisme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j:A\rightarrow N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j:A\rightarrow N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee974878f1fc194c215c00e59c8e4721df79f744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.343ex; height:2.509ex;" alt="{\displaystyle j:A\rightarrow N}" loading="lazy">, il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:K\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b1365bca781a49b19109ca08dd30956e5db6fa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.02ex; height:2.176ex;" alt="{\displaystyle h:K\rightarrow N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=h\circ i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcd481e9cf0f709666fcd418e4a0c1c792e562ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:8.419ex; height:2.509ex;" alt="{\displaystyle j=h\circ i}" loading="lazy">.

2. Si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30012178e674e69a55fee64e9c221dbdb0d95b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle M,N}" loading="lazy"> sont deux modèles de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> est sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> alors pour toute L-formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},...,x_{n},w)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},w)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a21328f9b43890ede0772cdd92e0272045fdbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.35ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},w)}" loading="lazy"> et tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},\ldots ,a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},\ldots ,a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e7dfb41e027e24804c6acd07429ef1158f29b60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.229ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},\ldots ,a_{n}}" loading="lazy"> dans le domaine de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, s'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> dans le domaine de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(a_{1},a_{2},...,a_{n},b)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a_{1},a_{2},...,a_{n},b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6ed0b7929c81c63796d86b5114794cac2374723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.802ex; height:2.843ex;" alt="{\displaystyle F(a_{1},a_{2},...,a_{n},b)}" loading="lazy"> est satisfaite dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, alors elle l'est aussi dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

Alors, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> admet l'élimination des quantificateurs.

<div class="mw-heading mw-heading4"><h4 id="Preuve_2">Preuve</h4></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f459119c1c9c030f07c7f4d7f24c5ed675e3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.841ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},y)}" loading="lazy"> une L-formule sans quantificateur. Supposons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30012178e674e69a55fee64e9c221dbdb0d95b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle M,N}" loading="lazy"> soient deux modèles de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> une sous-structure commune de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},...,a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},...,a_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56d574ff1f6ac680957241f7b55f7bf72ecde023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.22ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},...,a_{n}}" loading="lazy"> des éléments dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> un élément de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},a_{2},...,a_{n},b)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},a_{2},...,a_{n},b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52613b00259ba7f8e5e747c4bfb53b547ee54002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.549ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},a_{2},...,a_{n},b)}" loading="lazy">. Montrons qu'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models F(a_{1},a_{2},...,a_{n},c)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models F(a_{1},a_{2},...,a_{n},c)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a806035a05705a291ddcd84ecfbb36a6df894c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.18ex; height:2.843ex;" alt="{\displaystyle N\models F(a_{1},a_{2},...,a_{n},c)}" loading="lazy"> et puis conclure par le critère 1:

Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef8f6025f56d1b7fe6a9cf76678edcba0e8723d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.384ex; height:2.843ex;" alt="{\displaystyle M\models T}" loading="lazy"> et que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> est une sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, on a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\models T_{\forall }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\models T_{\forall }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab619c438c031a40c47eee36ac5f20195f7f19d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.552ex; height:2.843ex;" alt="{\displaystyle A\models T_{\forall }}" loading="lazy">. Par hypothèse 1) il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\models T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\models T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f8a59928a27817a8dc8d455d6bb87fe54ab6ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle K\models T}" loading="lazy"> telle que pour toute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\models T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q\models T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfd75712590b0661ed9cd25f823454d7bf2fee8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.78ex; height:2.843ex;" alt="{\displaystyle Q\models T}" loading="lazy"> qui est extension de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> est une sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy">. Par conséquent, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> est sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">.

Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},a_{2},...,a_{n},b),F}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},a_{2},...,a_{n},b),F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44628be6dd36fec367bff353fb2975f2f640fa5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.324ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},a_{2},...,a_{n},b),F}" loading="lazy"> est une formule sans quantificateur, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> est une sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> et que les formules sans quantificateurs sont préservées par la sous-structure, on a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\models \exists xF(a_{1},a_{2},...,a_{n},x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\models \exists xF(a_{1},a_{2},...,a_{n},x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa8120c8681a46a3c2541adc9dedc1b13066a6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.127ex; height:2.843ex;" alt="{\displaystyle K\models \exists xF(a_{1},a_{2},...,a_{n},x)}" loading="lazy">. De même <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models \exists xF(a_{1},a_{2},...,a_{n},x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models \exists xF(a_{1},a_{2},...,a_{n},x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88eba44e2313fff0e580cfda9e4e1b5f452a5463.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.125ex; height:2.843ex;" alt="{\displaystyle N\models \exists xF(a_{1},a_{2},...,a_{n},x)}" loading="lazy">.

Ainsi on conclut par le critère 2 que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> admet l'élimination des quantificateurs.

<div class="mw-heading mw-heading4"><h4 id="Exemple:_Corps_algébriquement_clos">Exemple: Corps algébriquement clos</h4></div>

On note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy"> pour la théorie des corps algébriquement clos. Pour démontrer que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy"> admet l'élimination des quantificateurs, on montre d'abord l'ensemble <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACFU}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10e9c0492a3c1ffc7b308c233514cb465010ded3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.033ex; height:2.176ex;" alt="{\displaystyle ACFU}" loading="lazy"> des conséquences universelles de la théorie des corps algébriquement clos est la <a href="Th%C3%A9orie_des_anneaux" title="Théorie des anneaux">théorie des anneaux</a> intègres. Et puis, on montre que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy"> vérifie les conditions du critère 3 pour conclure.

Montrons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACFU}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10e9c0492a3c1ffc7b308c233514cb465010ded3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.033ex; height:2.176ex;" alt="{\displaystyle ACFU}" loading="lazy"> est la théorie des anneaux intègres:

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> un <a href="Anneau_int%C3%A8gre" title="Anneau intègre">anneau intègre</a>. Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> corps-extension algébrique du corps de fraction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. On a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> est un modèle de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy">. Comme il existe un homomorphisme injectif de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> dans le corps de fraction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, et qu'il existe un homomorphisme injective du corps de fraction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">, on déduit que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> est un <a href="Sous-anneau" title="Sous-anneau">sous-anneau</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\models ACFU}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\models ACFU}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a4b0cf95df1985238c44c2d6f85446f809d1859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.081ex; height:2.843ex;" alt="{\displaystyle A\models ACFU}" loading="lazy">. Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\models ACFU}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\models ACFU}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc07fa84534f9071dd2ddfe5befaed4921b2f7c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.102ex; height:2.843ex;" alt="{\displaystyle B\models ACFU}" loading="lazy">. En particulier, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\models \forall x\forall y((x\not =0\land y\not =0)\rightarrow (x.y\not =0))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\models \forall x\forall y((x\not =0\land y\not =0)\rightarrow (x.y\not =0))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8318ce476c2402fbd9e70b3870af0d441afdfddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.551ex; height:2.843ex;" alt="{\displaystyle B\models \forall x\forall y((x\not =0\land y\not =0)\rightarrow (x.y\not =0))}" loading="lazy">. Comme de plus tous les axiomes de la théorie d'anneaux sont dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACFU}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10e9c0492a3c1ffc7b308c233514cb465010ded3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.033ex; height:2.176ex;" alt="{\displaystyle ACFU}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> est un anneau intègre.

Montrons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy"> vérifie la première condition du critère 3:

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> un modèle de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACFU}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10e9c0492a3c1ffc7b308c233514cb465010ded3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.033ex; height:2.176ex;" alt="{\displaystyle ACFU}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> est un anneau intègre. Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> le corps-extension algébrique du corps de fraction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">. Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> un modèle de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> est un sous-anneau de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">. Comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> est un corps contenant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> contenant le corps de fraction de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">. Et comme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> est algébriquement clos, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> contient, par définition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> le corps-extension algébrique de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">.

Montrons que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ACF}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a91dc2c9b0190ac5ae24797f90896bd62ee98c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.25ex; height:2.176ex;" alt="{\displaystyle ACF}" loading="lazy"> vérifie la deuxième condition du critère 3:

Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N\models ACF}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,N\models ACF}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/895dd46265d84c1b01dccd5e090a02af2be1b4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:2.843ex;" alt="{\displaystyle M,N\models ACF}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> est une sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f459119c1c9c030f07c7f4d7f24c5ed675e3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.841ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},y)}" loading="lazy"> une L-formule sans quantificateur, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},...,a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9bcc0e499b63bdf3e61e2769f2df6d70b2aff65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.902ex; height:2.009ex;" alt="{\displaystyle a_{1},...,a_{n}}" loading="lazy"> éléments du domaine de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Supposons qu'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> élément du domaine de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models F(a_{1},a_{2},...,a_{n},b)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models F(a_{1},a_{2},...,a_{n},b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f083fa46591abe96465f40430a048e938a894f62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.17ex; height:2.843ex;" alt="{\displaystyle N\models F(a_{1},a_{2},...,a_{n},b)}" loading="lazy">. Montrons qu'il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> élément du domaine de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},...,a_{n},c)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},...,a_{n},c)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e706d65e1273172cf8eddeecf724180e52b1ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.24ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},...,a_{n},c)}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f459119c1c9c030f07c7f4d7f24c5ed675e3665.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.841ex; height:2.843ex;" alt="{\displaystyle F(x_{1},...,x_{n},y)}" loading="lazy"> est une formule sans quantificateur donc elle est équivalente à une formule sous forme disjonctive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{ij}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigvee _{i}\bigwedge _{j}A_{ij}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c65e2dd29e43d3150c000c1e0c57caaa5f078165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.258ex; height:5.843ex;" alt="{\displaystyle \bigvee _{i}\bigwedge _{j}A_{ij}(x_{1},...,x_{n},y)}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{ij}(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{ij}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c96bb86558cf9a01455554e94f6a2aa37e2ad68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.321ex; height:3.009ex;" alt="{\displaystyle A_{ij}(x_{1},...,x_{n},y)}" loading="lazy"> sont des formules atomiques ou des négations de formules atomiques. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models F(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models F(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f0e2d5a9c49addefb9afd1b28192c702f3b61c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.21ex; height:2.843ex;" alt="{\displaystyle N\models F(x_{1},...,x_{n},y)}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models \bigwedge _{j}A_{ij}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models \bigwedge _{j}A_{ij}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/498ef07273349a6665b188f2f942600e793a11de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.658ex; height:5.843ex;" alt="{\displaystyle N\models \bigwedge _{j}A_{ij}(x_{1},...,x_{n},y)}" loading="lazy"> pour un certain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">. On peut donc supposer, <a href="Sans_perte_de_g%C3%A9n%C3%A9ralit%C3%A9" title="Sans perte de généralité">sans perte de généralité</a>, que F est une formule de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e4dc9488e69a6fcbb403946d9bb09846d42bffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.612ex; height:5.509ex;" alt="{\displaystyle \bigwedge _{i}A_{i}(x_{1},...,x_{n},y)}" loading="lazy"> où les <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ai(x_{1},...,x_{n},y)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Ai(x_{1},...,x_{n},y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/592ec53b892dfe4f6011d7d13e1aa75363bda4d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.646ex; height:2.843ex;" alt="{\displaystyle Ai(x_{1},...,x_{n},y)}" loading="lazy"> sont des formules atomiques ou des négations de formules atomiques. Et le langage de ACF est le langage d'anneau, une formule atomique <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x_{1},...,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(x_{1},...,x_{n})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8cb29eb7d69102450d97d3220aaabcbbce900d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.654ex; height:2.843ex;" alt="{\displaystyle A(x_{1},...,x_{n})}" loading="lazy"> est de la forme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{1},...,x_{n})=0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x_{1},...,x_{n})=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bb64d44924f60b94f69f2df5a70734e6818246e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.917ex; height:2.843ex;" alt="{\displaystyle P(x_{1},...,x_{n})=0}" loading="lazy"> où P est un <a href="Polyn%C3%B4me" title="Polynôme">polynôme</a> de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X_{1},X_{2},...,X_{n}]}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X_{1},X_{2},...,X_{n}]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8b3f0620e20d840be4ce83c78eda58577e058ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.147ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X_{1},X_{2},...,X_{n}]}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(a_{1},...,a_{n},X)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a_{1},...,a_{n},X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9300fa33eb9ec889a434eb308352afe418e4426c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.466ex; height:2.843ex;" alt="{\displaystyle F(a_{1},...,a_{n},X)}" loading="lazy"> est un polynôme de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy">. Donc il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1},P_{2},...,P_{n},Q_{1},Q_{2},...,Q_{m}}">

<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 P_{1},P_{2},...,P_{n},Q_{1},Q_{2},...,Q_{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e91e377001a2e6141187664d78355a1485ce3a3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.543ex; height:2.509ex;" alt="{\displaystyle P_{1},P_{2},...,P_{n},Q_{1},Q_{2},...,Q_{m}}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(a_{1},...,a_{n},X)=\bigwedge _{i=1}^{n}P_{i}(X)=0\wedge \bigwedge _{j=1}^{m}Q_{j}(X)\not =0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a_{1},...,a_{n},X)=\bigwedge _{i=1}^{n}P_{i}(X)=0\wedge \bigwedge _{j=1}^{m}Q_{j}(X)\not =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de855e73a7a0ac3e5e30ad9d088aadea75d67581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.508ex; height:7.176ex;" alt="{\displaystyle F(a_{1},...,a_{n},X)=\bigwedge _{i=1}^{n}P_{i}(X)=0\wedge \bigwedge _{j=1}^{m}Q_{j}(X)\not =0}" loading="lazy">. On a deux cas:

S'il existe un polynôme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c7bb3bc49771af415744e5dd40f5f93d14f5519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.581ex; height:2.509ex;" alt="{\displaystyle P_{k}}" loading="lazy"> non nul, alors comme on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models P_{i}(b)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models P_{i}(b)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8ef7a4f8df42132805cd37f9c9c479525548225.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.729ex; height:2.843ex;" alt="{\displaystyle N\models P_{i}(b)=0}" loading="lazy"> pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, on a en particulier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models P_{k}(b)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models P_{k}(b)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70112410b8e8df8e055c948a7365825af44ac997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.018ex; height:2.843ex;" alt="{\displaystyle N\models P_{k}(b)=0}" loading="lazy">. Comme M est un sous-corps algébriquement clos de N et b est un élément du domaine de N, on a que b est un élément du domaine de M.

Sinon, alors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(a_{1},...,a_{n},X)=\bigwedge _{j=1}^{m}Q_{j}(X)\not =0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a_{1},...,a_{n},X)=\bigwedge _{j=1}^{m}Q_{j}(X)\not =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c03541e6c7b88f47b50bb88adc6d33646b08d1ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.528ex; height:7.176ex;" alt="{\displaystyle F(a_{1},...,a_{n},X)=\bigwedge _{j=1}^{m}Q_{j}(X)\not =0}" loading="lazy">. Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/222db49df2eefdb67737ba2d2dbd221a1bae0bf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.335ex; height:2.843ex;" alt="{\displaystyle S_{j}}" loading="lazy"> l'ensemble de racines de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{j}(X)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{j}(X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9b2f63b73b22a90a6f649ac8fa00a67443b6cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.537ex; height:3.009ex;" alt="{\displaystyle Q_{j}(X)}" loading="lazy"> pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">. On sait que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/222db49df2eefdb67737ba2d2dbd221a1bae0bf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.335ex; height:2.843ex;" alt="{\displaystyle S_{j}}" loading="lazy"> est fini pour tout <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> et que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> est infini donc il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |M|-\bigcup _{j=1}^{m}S_{j}}">

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |M|-\bigcup _{j=1}^{m}S_{j}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8374b53122c9152d771cf63f9ed3389a6ed3181.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.076ex; height:7.176ex;" alt="{\displaystyle |M|-\bigcup _{j=1}^{m}S_{j}}" loading="lazy"> tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge _{j=1}^{m}Q_{j}(c)\not =0}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigwedge _{j=1}^{m}Q_{j}(c)\not =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f761d996f5971b0b9ffe78d71008edbee0a748f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.019ex; width:13.009ex; height:7.176ex;" alt="{\displaystyle \bigwedge _{j=1}^{m}Q_{j}(c)\not =0}" loading="lazy">. Donc il existe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> dans le domaine de M tel que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F(a_{1},...,a_{n},c)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F(a_{1},...,a_{n},c)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e706d65e1273172cf8eddeecf724180e52b1ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.24ex; height:2.843ex;" alt="{\displaystyle M\models F(a_{1},...,a_{n},c)}" loading="lazy">.

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

Exemples de théories admettant l'élimination des quantificateurs :

<ul><li><a href="Corps_r%C3%A9el_clos" title="Corps réel clos">corps réels clos</a> (théorème de Tarski-Seidenberg <a href="https://en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem" class="extiw external" title="en:Tarski–Seidenberg theorem">(en)</a>, algorithme de décomposition algébrique cylindrique <a href="https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition" class="extiw external" title="en:Cylindrical algebraic decomposition">(en)</a>) ;</li>

<li><a href="Corps_alg%C3%A9briquement_clos" title="Corps algébriquement clos">corps algébriquement clos</a> de <a href="Caract%C3%A9ristique_d'un_anneau" title="Caractéristique d'un anneau">caractéristique</a> quelconque ou fixée<a href="#cite_note-1">[1]</a> (par exemple via les <a href="Base_de_Gr%C3%B6bner" title="Base de Gröbner">bases de Gröbner</a> et le théorème d'extension) ;</li>

<li>théorie linéaire des rationnels ou des réels (algorithmes de <a href="Jeanne_Ferrante" title="Jeanne Ferrante">Ferrante</a> et <a href="Charles_Rackoff" title="Charles Rackoff">Rackoff</a><a href="#cite_note-2">[2]</a>,<a href="#cite_note-3">[3]</a>, de Loos et Weispfenning <a href="https://de.wikipedia.org/wiki/Volker_Weispfenning" class="extiw external" title="de:Volker Weispfenning">(de)</a><a href="#cite_note-4">[4]</a>) ;</li>

<li>théorie linéaire des entiers, dite aussi <a href="Arithm%C3%A9tique_de_Presburger" title="Arithmétique de Presburger">arithmétique de Presburger</a>.</li></ul>

Toutes ces théories sont donc <a href="Th%C3%A9orie_mod%C3%A8le-compl%C3%A8te" title="Théorie modèle-complète">modèle-complètes</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Modèle-complétude">Modèle-complétude</h3></div>

Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> une <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">-théorie admettant l'élimination des quantificateurs. Soient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30012178e674e69a55fee64e9c221dbdb0d95b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle M,N}" loading="lazy"> deux modèles de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> tels que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> est une sous-structure de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. Soit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> une <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">-formule et <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> une assignation des variables dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Par l'élimination des quantificateurs, il existe une <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">-formule sans quantificateur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> qui est équivalente à <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">. On a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fe03b3151760362ca8aae9bc74ba9fa2a975ccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.872ex; height:2.843ex;" alt="{\displaystyle M\models F[s]}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models G[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models G[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23a9330eb4cb89f0b491e02abaa31b5f0f7500f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.958ex; height:2.843ex;" alt="{\displaystyle M\models G[s]}" loading="lazy"> et que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models F[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models F[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da6b41e19aa0e29d40a1922f047b1e2feeffa236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.494ex; height:2.843ex;" alt="{\displaystyle N\models F[s]}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models G[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models G[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f7939d1f3563eb3b20eedf625642965fbaf61ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.58ex; height:2.843ex;" alt="{\displaystyle N\models G[s]}" loading="lazy">. Comme l'injection canonique de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> dans <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> est un homomorphisme injectif et que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> est une formule sans quantificateur, on a que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models G[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models G[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23a9330eb4cb89f0b491e02abaa31b5f0f7500f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.958ex; height:2.843ex;" alt="{\displaystyle M\models G[s]}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models G[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models G[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f7939d1f3563eb3b20eedf625642965fbaf61ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.58ex; height:2.843ex;" alt="{\displaystyle N\models G[s]}" loading="lazy">. On conclut que <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models F[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models F[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fe03b3151760362ca8aae9bc74ba9fa2a975ccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.872ex; height:2.843ex;" alt="{\displaystyle M\models F[s]}" loading="lazy"> si et seulement si <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\models F[s]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\models F[s]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da6b41e19aa0e29d40a1922f047b1e2feeffa236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.494ex; height:2.843ex;" alt="{\displaystyle N\models F[s]}" loading="lazy">. Donc <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> est une sous-structure élémentaire de <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. Ainsi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> est modèle-complète (par définition).

<div class="mw-heading mw-heading2"><h2 id="Notes_et_références">Notes et références</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Philipp Rothmaler, <cite class="italique" lang="en">Introduction to Model Theory</cite>, <a href="CRC_Press" title="CRC Press">CRC Press</a>, <time>2000</time> (<a rel="nofollow" class="external text" href="https://books.google.fr/books?id=0_NNAR6ztIUC&pg=PA141">lire en ligne</a>), <abbr class="abbr" title="page">p.</abbr> 141<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Model+Theory&rft.pub=CRC+Press&rft.aulast=Rothmaler&rft.aufirst=Philipp&rft.date=2000&rft.pages=141&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89limination+des+quantificateurs">.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Jeanne Ferrante et Charles Rackoff, « <cite style="font-style:normal" lang="en">A decision procedure for the first order theory of real addition with order</cite> », <a href="Liste_des_journaux_scientifiques_en_math%C3%A9matiques#S" title="Liste des journaux scientifiques en mathématiques">SIAM J. on Computing</a>, <abbr class="abbr" title="volume">vol.</abbr> 4, <abbr class="abbr" title="numéro">no</abbr> 1,‎ <time class="nowrap" datetime="1975-03" data-sort-value="1975-03">mars 1975</time>, <abbr class="abbr" title="pages">p.</abbr> 69-76 (<a href="Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1137/0204006">10.1137/0204006</a>)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+decision+procedure+for+the+first+order+theory+of+real+addition+with+order&rft.jtitle=SIAM+J.+on+Computing&rft.issue=1&rft.aulast=Ferrante&rft.aufirst=Jeanne&rft.au=Rackoff%2C+Charles&rft.date=1975-03&rft.volume=4&rft.pages=69-76&rft_id=info%3Adoi%2F10.1137%2F0204006&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89limination+des+quantificateurs">.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Aaron R. Bradley et <a href="Zohar_Manna" title="Zohar Manna">Zohar Manna</a>, <cite class="italique" lang="en">The Calculus of Computation : Decision Procedures with Applications to Verification</cite>, Berlin, <a href="Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer</a>, <time class="nowrap" datetime="2007-10" data-sort-value="2007-10">octobre 2007</time>, 366 <abbr class="abbr" title="pages">p.</abbr> (<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> 978-3-540-74112-1)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Calculus+of+Computation+%3A+Decision+Procedures+with+Applications+to+Verification&rft.place=Berlin&rft.pub=Springer&rft.aulast=Bradley&rft.aufirst=Aaron+R.&rft.au=Manna%2C+Zohar&rft.date=2007-10&rft.tpages=366&rft.isbn=978-3-540-74112-1&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89limination+des+quantificateurs">.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Rüdiger Loos et Volker Weispfenning, « <cite style="font-style:normal" lang="en">Applying linear quantifier elimination</cite> », <a href="The_Computer_Journal" title="The Computer Journal">The Computer Journal</a>, <abbr class="abbr" title="volume">vol.</abbr> 36, <abbr class="abbr" title="numéro">no</abbr> 5,‎ <time>1993</time>, <abbr class="abbr" title="pages">p.</abbr> 450-462 (<a href="Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1093/comjnl/36.5.450">10.1093/comjnl/36.5.450</a>, <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.8844&rep=rep1&type=pdf">lire en ligne</a> <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>)<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Applying+linear+quantifier+elimination&rft.jtitle=The+Computer+Journal&rft.issue=5&rft.aulast=Loos&rft.aufirst=R%C3%BCdiger&rft.au=Weispfenning%2C+Volker&rft.date=1993&rft.volume=36&rft.pages=450-462&rft_id=info%3Adoi%2F10.1093%2Fcomjnl%2F36.5.450&rfr_id=info%3Asid%2Ffr.wikipedia.org%3A%C3%89limination+des+quantificateurs">.

</li>

</ol></div>

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<div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3></div>

<ul><li><a href="Skol%C3%A9misation" title="Skolémisation">Skolémisation</a></li>

<li><a href="Th%C3%A9or%C3%A8me_de_Herbrand" title="Théorème de Herbrand">Théorème de Herbrand</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Lien_externe">Lien externe</h3></div>

<li>Francois Schwarzentruber, « <a rel="nofollow" class="external text" href="http://www.irit.fr/~Francois.Schwarzentruber/realqelim/"><cite style="font-style:normal; color:var(--color-link-red, #d73333);">Explication d'un algorithme d'éliminations des quantificateurs</cite></a> »(<a rel="nofollow" class="external text" href="https://web.archive.org/web/*/http://www.irit.fr/~Francois.Schwarzentruber/realqelim/">Archive.org</a> • <a rel="nofollow" class="external text" href="https://archive.wikiwix.com/cache/?url=http://www.irit.fr/~Francois.Schwarzentruber/realqelim/">Wikiwix</a> • <a rel="nofollow" class="external text" href="https://archive.is/http://www.irit.fr/~Francois.Schwarzentruber/realqelim/">Archive.is</a> • <a rel="nofollow" class="external text" href="https://webcache.googleusercontent.com/search?hl=fr&q=cache:http://www.irit.fr/~Francois.Schwarzentruber/realqelim/">Google</a> • Que faire ?)</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3></div>

<ul><li><a href="Jean-Louis_Krivine" title="Jean-Louis Krivine">Jean-Louis Krivine</a> et <a href="Georg_Kreisel" title="Georg Kreisel">Georg Kreisel</a>, Éléments de logique mathématique (théorie des modèles), Paris, <a href="%C3%89ditions_Dunod" title="Éditions Dunod">Dunod</a>, 1966, chap. 4, <a rel="nofollow" class="external text" href="http://www.pps.jussieu.fr/~krivine/articles/Elements_logique.pdf">pdf</a></li>

<li>Jean-François Pabion, Logique mathématique, chapitre VI « Élimination des quantificateurs » pp. 191-210, Hermann, Paris 1976 (<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> 2-7056 5830-0)</li>

<li>René David, Karim Nour, Christophe Raffalli, Introduction à la logique-theorie de la démonstration, 2e édition, Dunod</li>

<li>David Marker, Model Theory: An introduction, Springer</li>

<li><a href="Ren%C3%A9_Cori" title="René Cori">René Cori</a>, Daniel Lascar, Logique mathématique 2, Dunod</li></ul>

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