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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#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="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-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">p</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">a</font> be  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>









<b>set </b><font color="Maroon">ht</font> =  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">htp</font> =  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>;<br/>
<a NAME="E1:34"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</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="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
(  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <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="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green">E24</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <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="34_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">sp</font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</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">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">sp</font>;<br/><a NAME="E4:34_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E6:34_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</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="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E7:34_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</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="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><a NAME="E8:34_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>;<br/><a NAME="E9:34_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</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="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><b>hence </b><a NAME="E10:34_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><b>hence </b> <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> = 
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>
<br/>.= 
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_2_1"/><i><font color="Green">E49</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon">sp</font> =  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">sp</font>;<br/><a NAME="E4:34_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><a NAME="E6:34_1_2_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</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="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E7:34_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>;<br/><a NAME="E8:34_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</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="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>;<br/><b>hence </b><a NAME="E9:34_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><a NAME="E3:34_1_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E4:34_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</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#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="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E5:34_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</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">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>;<br/><a NAME="E6:34_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</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#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="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a>, , <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><a NAME="E7:34_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<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">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2"><b>now </b></a><div class="add"><b>let </b><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="E1:34_1_2_2"/><i><font color="Green">E53</font></i>: 
<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">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E2:34_1_2_2"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:34_1_2_2"/>
<font color="Maroon">b</font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:34_1_2"/>
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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