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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">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  <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> <a href="algstr_1.html#V1">left_zeroed</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">P</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>let </b><font color="Maroon">f'</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="E1:80"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1"/>
( <font color="Maroon">h'</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> or <font color="Maroon">h'</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><b>suppose </b></a><a NAME="E1:80_1_1"/>
<font color="Maroon">h'</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:80_1_1"/><b>then </b>
<font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:80_1_1"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</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="rewrite1.html#T13">REWRITE1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2"><b>suppose </b></a><a NAME="E1:80_1_2"/>
<font color="Maroon">h'</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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h'</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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><b>set </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">g</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1_2_1_1"/>
( <font color="Maroon">f'</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2_1_1_1"><b>case </b></a><a NAME="E1:80_1_2_1_1_1"/>
<font color="Maroon">f'</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:80_1_2_1_1_1"/><b>then </b>
<font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font> <a href="funct_2.html#R2">=</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#T87">POLYNOM1:87</a></i>;<br/><b>hence </b><a NAME="E3:80_1_2_1_1_1"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</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="rewrite1.html#T13">REWRITE1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2_1_1_2"><b>case </b></a><a NAME="E1:80_1_2_1_1_2"/>
<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>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f'</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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><b>assume </b><a NAME="E3:80_1_2_1_1_2"/>
not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><a NAME="E4:80_1_2_1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">g</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_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="E1:80_1_2_1_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">g</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">g'</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="E3:80_1_2_1_1_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> &amp; not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">g'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 ;<br/><b>thus </b><a NAME="E4:80_1_2_1_1_2_1"/>
<font color="Maroon">u</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="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> : not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">g</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> </span>}</span>  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> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:80_1_2_1_1_2_2"/>
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">m</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="E4:80_1_2_1_1_2_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">m'</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">m'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">m</font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Olive">m'</font>,<font color="Maroon">T</font> ) )
 <b>by </b><i/>;<br/><a NAME="E5:80_1_2_1_1_2_2"/>
<font color="Maroon">m</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1_2_1_1_2_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">m</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>

<b>consider </b><font color="Maroon">g'</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="E3:80_1_2_1_1_2_2_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">m</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g'</font> &amp; not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">g'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:80_1_2_1_1_2_2_1"/>
<font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:80_1_2_1_1_2_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rewrite1.html#T13">REWRITE1:13</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E7:80_1_2_1_1_2_2"/><i><font color="Green">E50</font></i>: 
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="termord.html#T31">TERMORD:31</a></i>;<br/><a NAME="E8:80_1_2_1_1_2_2"/>
<font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E9:80_1_2_1_1_2_2"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E10:80_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><a NAME="E11:80_1_2_1_1_2_2"/>
( <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> &amp; <font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E12:80_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font> <a href="polyred.html#R3" target="_self">reduces_to</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">f</font>, <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E13:80_1_2_1_1_2_2"/>
<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="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E14:80_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="termord.html#T17">TERMORD:17</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#T32">TERMORD:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23">VECTSP_1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="termord.html#D8">TERMORD:def 8</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K29">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="polynom1.html#T85">POLYNOM1:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K25">-</a> <span class="p2">(<span class="default"><a href="termord.html#K5">HM</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="termord.html#D9">TERMORD:def 9</a></i>
;<br/><a NAME="E17:80_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font> <a href="polyred.html#R4" target="_self">reduces_to</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>,<font color="Maroon">f</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:80_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font> <a href="polyred.html#R5" target="_self">reduces_to</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>,<font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="txt" href="polyred.html#E7:80_1_2_1_1_2_2"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E19:80_1_2_1_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E20:80_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">m</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T16">REWRITE1:16</a></i>;<br/><a NAME="E21:80_1_2_1_1_2_2"/>
 <a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font> <a href="polyred.html#R2" target="_self">&lt;</a> <font color="Maroon">m</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/><a NAME="E22:80_1_2_1_1_2_2"/><b>then </b>
not <font color="Maroon">m</font> <a href="polyred.html#R1" target="_self">&lt;=</a>  <a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3"><i><font color="Green">Lemma39</font></i></a></i>;<br/><a NAME="E23:80_1_2_1_1_2_2"/><b>then </b>
not  <a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><a NAME="E24:80_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">m</font>,<font color="Maroon">T</font></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><b>consider </b><font color="Maroon">u</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="E26:80_1_2_1_1_2_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">m</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">u</font> &amp; not  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">u</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E27:80_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rewrite1.html#T17">REWRITE1:17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:80_1_2_1_1_2"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <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></div><b>end;</b></div><b>hence </b><a NAME="E4:80_1_2"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">h'</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">f'</font>, <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>

</div>
