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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> 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>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green">E23</font></i>: 
<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>
 ;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">T</font> #);<br/>
<b>assume </b><a NAME="E2:53"/><i><font color="Green">E24</font></i>: 
(  <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font> &amp;  <a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="polyred.html#R1" target="_self">&lt;=</a>  <a href="termord.html#K6">Red</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>,<font color="Maroon">T</font> )
 ;<br/>

<a NAME="E3:53"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13">FinOrd</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">T</font> #)
 
<b>by </b><i/>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12">FinOrd-Approx</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">T</font> #)</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bagorder.html#D17">BAGORDER:def 17</a></i>;<br/>
<b>set </b><font color="Maroon">rp</font> =  <a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">rq</font> =  <a href="termord.html#K6">Red</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">y</font> =  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>case </b></a><a NAME="E1:53_1_1_1"/>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1"/>
<font color="Maroon">p</font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>case </b></a><a NAME="E1:53_1_1_2"/><i><font color="Green">E27</font></i>: 
<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>
 ;<br/><div class="add"><a NAME="E2:53_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font> is <a href="polynom7.html#V1">non-zero</a>
 <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E3:53_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">q</font> is <a href="polynom7.html#V1">non-zero</a>
 <b>by </b><i>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E4:53_1_1_2"/><i><font color="Green">E50</font></i>: 
(  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</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/><i><font color="Green">E51</font></i>:  <a href="bagorder.html#K11">PosetMax</a> <span class="p1">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> = 
 <a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="bagorder.html#K11">PosetMax</a> <span class="p1">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E3:53_1_1_2"><i><font color="Green">E49</font></i></a>, </i>
;<br/><i><font color="Green">E52</font></i>:  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="termord.html#T36">TERMORD:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11">PosetMax</a> <span class="p3">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, </i>
;<br/> <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="termord.html#K6">Red</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="termord.html#T36">TERMORD:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11">PosetMax</a> <span class="p3">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="polyred.html#E3:53_1_1_2"><i><font color="Green">E49</font></i></a>, </i>
;<br/><a NAME="E8:53_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12">FinOrd-Approx</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">T</font> #)</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="polyred.html#E4:53_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="polyred.html#E5:53_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="polyred.html#E6:53_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="bagorder.html#T36">BAGORDER:36</a></i>;<br/><a NAME="E9:53_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">q</font>,<font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13">FinOrd</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">T</font> #)
 <b>by </b><i><a class="ref" href="bagorder.html#D17">BAGORDER:def 17</a></i>;<br/><b>hence </b><a NAME="E10:53_1_1_2"/>
<font color="Maroon">p</font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<font color="Maroon">p</font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">q</font>,<font color="Maroon">T</font>
 ;<br/>


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