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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> <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>;<br/><b>let </b><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>let </b><font color="Maroon">m</font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>











<b>assume </b><a NAME="E1:36"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<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="E2:36"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</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">a</font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="polynom7.html#K2">term</a> <font color="Maroon">m</font>;<br/>
<a NAME="E3:36"/>
<font color="Maroon">m</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="E4:36"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="termord.html#K4">HC</a> <font color="Maroon">m</font>,<font color="Maroon">T</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="termord.html#T17">TERMORD:17</a></i>;<br/>
<a NAME="E5:36"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">m</font>,<font color="Maroon">T</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="termord.html#D7">TERMORD:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">m</font> as  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="termord.html#T23">TERMORD:23</a></i>;<br/>
<a NAME="E7:36"/>
<font color="Maroon">m</font> <a href="funct_2.html#R2">=</a>  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>;<br/>
<a NAME="E8:36"/><b>then </b>
<font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<b>then </b> <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> = 
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:36"><i><font color="Green">E24</font></i></a></i>
;<br/>
<a NAME="E10:36"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</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"><a href="termord.html#K3">HT</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</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">HT</a> <font color="Maroon">p</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:36"/>
<font color="Maroon">m</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">m</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="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>;<br/>
<a NAME="E12:36"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</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"><a href="termord.html#K3">HT</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</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="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:36"/>
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


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