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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  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">m</font> be   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st  <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="ref" href="polynom7.html#T6">POLYNOM7:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><i><font color="Green">E33</font></i>:  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font> = 
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>
<br/>.= 
 <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">m</font>,<font color="Maroon">O</font>
<b>by </b><i>, </i>
;<br/><br/><b>hence </b><a NAME="E3:39_1_1"/>
 <a href="termord.html#K4" target="_self">HC</a> <font color="Maroon">m</font>,<font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>;<br/><b>hence </b><a NAME="E4:39_1_1"/>
 <a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">m</font>,<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="termord.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green">E35</font></i>: 
 ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st  <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:39_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E4:39_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="termord.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:39_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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="E6:39_1_2"/>
 <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">m</font>,<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><i><font color="Green">E44</font></i>:  <a href="termord.html#K3" target="_self">HT</a> <font color="Maroon">m</font>,<font color="Maroon">O</font> = 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="termord.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>
;<br/><br/><b>hence </b><a NAME="E8:39_1_2"/>
 <a href="termord.html#K4" target="_self">HC</a> <font color="Maroon">m</font>,<font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>;<br/><b>hence </b><a NAME="E9:39_1_2"/>
 <a href="termord.html#K5" target="_self">HM</a> <font color="Maroon">m</font>,<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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