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<div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<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">g</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p</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="E1:56"/>
<font color="Maroon">f</font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">T</font>
 ;<br/>

<b>then 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="E3:56"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D6">POLYRED:def 6</a></i>;<br/>
<a NAME="E4:56"/><i><font color="Green">E40</font></i>: 
( <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">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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">s</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">s</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:56"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1"/>
 <a href="polynom1.html#K24">-</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>
 ;<br/><b>then </b><font color="Maroon">f</font> = 
 <a href="polynom1.html#K24">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>

<br/>.= 
 <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="E3:56_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:56"/>
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/>
<b>then </b> <a href="polynom1.html#K24">-</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</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"><a href="polynom1.html#K24">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T1">POLYRED:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K25">-</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"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></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"><a href="polynom1.html#K24">-</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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T9">POLYRED:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</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"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">p</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="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></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"><a href="polynom1.html#K24">-</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"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span></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">p</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="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</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"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></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"><a href="polynom1.html#K24">-</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"><a href="polynom1.html#K24">-</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D22">POLYNOM1:def 22</a></i>
;<br/>
<a NAME="E10:56"/><b>then </b>
(  <a href="polynom1.html#K24">-</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> &amp; <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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">f</font></span>)</span> &amp;  ex <font color="Olive">s</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">s</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="polynom1.html#K24">-</a> <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</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"><a href="polynom1.html#K24">-</a> <font color="Maroon">f</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></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">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="polyred.html#K1">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> ) )
 
<b>by </b><i>, , , <a class="ref" href="groeb_1.html#T5">GROEB_1:5</a></i>;<br/>
<a NAME="E11:56"/><b>then </b>
 <a href="polynom1.html#K24">-</a> <font color="Maroon">f</font> <a href="polyred.html#R3">reduces_to</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:56"/>
 <a href="polynom1.html#K24">-</a> <font color="Maroon">f</font> <a href="polyred.html#R4">reduces_to</a>  <a href="polynom1.html#K24">-</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D6">POLYRED:def 6</a></i>;<br/>


</div>
