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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">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">G</font> be   <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="E1:36"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">G</font> <a href="groeb_1.html#R1">is_Groebner_basis_wrt</a> <font color="Maroon">T</font>
 ;<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="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_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="36_1_1"><b>suppose </b></a><a NAME="E1:36_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="E2:36_1_1"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font>, <font color="Olive">h</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">h</font> <a href="rewrite1.html#R4">is_a_normal_form_of</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Olive">g1</font>,<font color="Olive">g2</font>,<font color="Maroon">T</font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><a NAME="E1:36_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">G</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="E3:36_1_2"/>
 <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>, <a class="ref" href="groeb_1.html#D3">GROEB_1:def 3</a></i>;<br/><a NAME="E4:36_1_2"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V7">confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T12">GROEB_1:12</a></i>;<br/><a NAME="E5:36_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V4">with_UN_property</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T13">GROEB_1:13</a></i>;<br/><a NAME="E6:36_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V8">with_Church-Rosser_property</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T14">GROEB_1:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g1</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">g2</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">h</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:36_1_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">h</font> <a href="rewrite1.html#R4">is_a_normal_form_of</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> )
 ;<br/><a NAME="E2:36_1_2_1"/><b>then </b>
 <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</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/><a NAME="E3:36_1_2_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font>, <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="groeb_1.html#T15">GROEB_1:15</a></i>;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_1_1"/>
not  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="rewrite1.html#R3">is_a_normal_form_wrt</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">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_2_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</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>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <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="ref" href="rewrite1.html#D5">REWRITE1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">f1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">f2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_1_2_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f1</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">{</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">}</a></span> &amp; <font color="Maroon">f2</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; <span class="p1"><a href="tarski.html#K4">[</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>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</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/><a NAME="E6:36_1_2_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f1</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; not <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1">{</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">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><font color="Maroon">f1</font> = 
<span class="p1"><a href="tarski.html#K4">[</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>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
 <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="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E8:36_1_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E5:36_1_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">h</font> <a href="rewrite1.html#R3">is_a_normal_form_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font>,<font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D6">REWRITE1:def 6</a></i>;<br/><a NAME="E6:36_1_2_1"/><b>then </b>
(  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">h</font>, <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">g1</font>,<font color="Maroon">g2</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="rewrite1.html#T26">REWRITE1:26</a></i>;<br/><a NAME="E7:36_1_2_1"/><b>then </b>
<font color="Maroon">h</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T31">REWRITE1:31</a></i>;<br/><b>hence </b><a NAME="E8:36_1_2_1"/>
<font color="Maroon">h</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="groeb_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="rewrite1.html#D19">REWRITE1:def 19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:36_1_2"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font>, <font color="Olive">h</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">h</font> <a href="rewrite1.html#R4">is_a_normal_form_of</a>  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Olive">g1</font>,<font color="Olive">g2</font>,<font color="Maroon">T</font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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