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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">O</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="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-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="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">q</font> be  <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>;<br/>







<a NAME="E1:50"/>
( <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">q</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="E2:50"/><b>then </b>
(  <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>  &amp;  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> <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="E3:50"/><b>then </b>
(  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp;  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:50"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</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> &amp; <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</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/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font></span>)</span>;<br/>
<a NAME="E5:50"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font></span>)</span> 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/>
<b>assume </b><a NAME="E6:50"/>
not <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>

<a NAME="E7:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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="E8:50"/><b>then </b>
<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" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:50"/>
contradiction
 <b>by </b><i><a class="txt" href="termord.html#E5:50"><i><font color="Green">E33</font></i></a>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>


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