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<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#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="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">p</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>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b'</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>









<b>assume </b><a NAME="E1:73"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">b</font> <a href="termord.html#R2">&lt;</a> <font color="Maroon">b'</font>,<font color="Maroon">T</font>
 ;<br/>

<b>assume </b><a NAME="E2:73"/>
<font color="Maroon">f</font> <a href="polyred.html#R3" target="_self">reduces_to</a> <font color="Maroon">g</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 ;<br/>

<b>then 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="E4:73"/><i><font color="Green">E24</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E5:73"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b'</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1"/>
<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"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E2:73_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">b'</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:73_1"/>
<font color="Maroon">b</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b'</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#D3">TERMORD:def 3</a></i>;<br/><a NAME="E4:73_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><b>hence </b><a NAME="E5:73_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="termord.html#D3">TERMORD:def 3</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E49</font></i>: <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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K25">-</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K22">+</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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<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#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><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" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D22">POLYNOM1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E9:73"/>
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
 <b>by </b><i/>;<br/>


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