<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V2">left-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:52"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">G</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E3:52"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:52"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">G</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V9">locally-confluent</a> )
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
(  not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> or  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/>
not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1"/>
 ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> st <br/>( <font color="Olive">G</font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font> &amp; not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>;<br/><b>set </b><font color="Maroon">R'</font> =  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span></span>)</span>,<font color="Maroon">T</font>;<br/><a NAME="E2:52_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:52_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span></span>)</span>,<font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/><a NAME="E5:52_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span></span>)</span>,<font color="Maroon">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_1_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_2_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E5:52_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E6:52_1_2_1"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:52_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E5:52_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E9:52_1_2_1"/>
<font color="Maroon">p</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">G</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E2:52_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:52_1_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">p</font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E13:52_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">p</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:52_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="polyred.html#D6">POLYRED:def 6</a></i>;<br/>
<a NAME="E14:52_1_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<a NAME="E15:52_1_2_1"/><b>then </b>
not <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:52_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:52_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E17:52_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">q</font>,<font color="Maroon">G'</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:52_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/>
<b>hence </b><a NAME="E18:52_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span></span>)</span>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polyred.html#D13">POLYRED:def 13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:52_1_2"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span></span>)</span>,<font color="Maroon">T</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:52"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E7:52_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span></span>)</span>,<font color="Maroon">T</font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="groeb_1.html#E1:52_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_2_1"/>
( <font color="Maroon">G'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2_1_1"><b>case </b></a><a NAME="E1:52_1_2_2_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">G'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_2_1_1_1_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:52_1_2_2_1_1_1_1_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_2_2_1_1_1_1_1"/>
<font color="Maroon">G'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:52_1_2_2_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_2_1_1_1_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:52_1_2_2_1_1_1_1_2_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_2_1_1_1_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_2_1_1_1_1_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 ;<br/><a NAME="E2:52_1_2_2_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:52_1_2_2_1_1_1_1_2_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_2_1_1_1_1_2_2_1"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:52_1_2_2_1_1_1_1_2_2_1"/><b>then </b>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_2_1_1_1_1_2_2_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:52_1_2_2_1_1_1_1_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:52_1_2"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:52_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E5:52_1_2_2_1_1_1_1_2"/>
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E6:52_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> is   <a href="ideal_1.html#NM1">Ideal</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T7">IDEAL_1:7</a></i>;<br/><b>hence </b><a NAME="E7:52_1_2_2_1_1_1_1_2"/>
<font color="Maroon">G'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E1:52"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="ideal_1.html#T44">IDEAL_1:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_2_2_1_1"/>
<font color="Maroon">G'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2_1_2"><b>case </b></a><a NAME="E1:52_1_2_2_1_2"/>
<font color="Maroon">G'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">GG</font> = <font color="Maroon">G</font>, <font color="Maroon">GG'</font> = <font color="Maroon">G'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> ;<br/><a NAME="E3:52_1_2_2_1_2"/><i><font color="Green">E93</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_2_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><b>then consider </b><font color="Maroon">f</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">GG</font><b> such that </b><br/><a NAME="E3:52_1_2_2_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">GG'</font><b> such that </b><br/><a NAME="E5:52_1_2_2_1_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a>, , </i>;<br/><b>thus </b><a NAME="E6:52_1_2_2_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_2_1_2_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><a NAME="E2:52_1_2_2_1_2_2"/>
<font color="Maroon">GG'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">GG</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ideal_1.html#T57">IDEAL_1:57</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_2_1_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:52_1_2_2_1_2"/>
<font color="Maroon">G'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="groeb_1.html#E5:52_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:52_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">G'</font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:52_1_2"><i><font color="Green">E45</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_3"/>
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G'</font>
 ;<br/><a NAME="E2:52_1_2_3"/><b>then </b>
not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:52_1_2"/>
 ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> st <br/>( <font color="Olive">G</font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font> &amp; not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:52_1_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
