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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#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> non <a href="realset2.html#V3">trivial</a>  <a href="rlvect_1.html#L1">LoopStr</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/>







<a NAME="E1:64"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><b>case </b></a><a NAME="E1:64_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <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="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <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="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/>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_2"><b>case </b></a><a NAME="E1:64_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</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"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:64_1_1_1_2"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:64_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p2">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
<span class="p1">(<span class="default"><a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p1">(<span class="default"><a href="termord.html#K6" target="_self">Red</a> <font color="Maroon">p</font>,<font color="Maroon">O</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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