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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> 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="E1:52"/>
 <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="termord.html#E2:52"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E2:52"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="termord.html#K3" target="_self">HT</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>,<font color="Maroon">O</font> <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>
 
<b>by </b><i/>;<br/>
<a NAME="E3:52"/>
<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>
 
<b>by </b><i><a class="txt" href="termord.html#E2:52"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E4:52"/><b>then </b>
<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="termord.html#R1" target="_self">&lt;=</a>  <a href="termord.html#K3" target="_self">HT</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>,<font color="Maroon">O</font>,<font color="Maroon">O</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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>,<span class="p2">(<span class="default"><a href="termord.html#K3" target="_self">HT</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">O</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:52"/>
 <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> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="polynom1.html#K9">+</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">s</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>, <font color="Olive">t</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="termord.html#E5:52"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E7:52"/><b>then </b>
 <a href="termord.html#K3" target="_self">HT</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>,<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="polynom1.html#K9">+</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">s</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>, <font color="Olive">t</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> : ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="termord.html#E9:52"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">t</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E9:52"/><i><font color="Green">E35</font></i>: 
(  <a href="termord.html#K3" target="_self">HT</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>,<font color="Maroon">O</font> <a href="pboole.html#R6">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">t</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font> )
 ;<br/>
<a NAME="E10:52"/>
( <font color="Maroon">s</font> <a href="termord.html#R1" target="_self">&lt;=</a>  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">p</font>,<font color="Maroon">O</font>,<font color="Maroon">O</font> &amp; <font color="Maroon">t</font> <a href="termord.html#R1" target="_self">&lt;=</a>  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">q</font>,<font color="Maroon">O</font>,<font color="Maroon">O</font> )
 
<b>by </b><i><a class="txt" href="termord.html#E11:52"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<a NAME="E11:52"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<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><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">t</font>,<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><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:52"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">t</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <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> <font color="Maroon">t</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp;  <a href="termord.html#K3" target="_self">HT</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>,<font color="Maroon">O</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <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/>
<a NAME="E13:52"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">t</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="bagorder.html#D7">BAGORDER:def 7</a></i>;<br/>
<a NAME="E14:52"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="polynom1.html#K9">+</a> <span class="p3">(<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>,<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="bagorder.html#D7">BAGORDER:def 7</a></i>;<br/>
<a NAME="E15:52"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">t</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="txt" href="termord.html#E12:52"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="orders_1.html#T14">ORDERS_1:14</a></i>;<br/>
<b>hence </b><a NAME="E16:52"/>
 <a href="termord.html#K3" target="_self">HT</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>,<font color="Maroon">O</font> <a href="pboole.html#R6">=</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="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>
 <b>by </b><i>, <a class="txt" href="termord.html#E11:52"><i><font color="Green">E37</font></i></a>, <a class="txt" href="termord.html#E12:52"><i><font color="Green">E39</font></i></a>, <a class="ref" href="orders_1.html#T13">ORDERS_1:13</a></i>;<br/>


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