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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>











<b>assume </b><a NAME="E1:36"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E3:36"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">E31</font></i>: 
 <a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="polynom7.html#K3">coefficient</a> <font color="Maroon">c<sub>6</sub></font> as  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polyred.html#E4:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="termord.html#T23">TERMORD:23</a></i>;<br/>
<a NAME="E7:36"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a>  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polyred.html#T22" target="_self">Th22</a></i>;<br/>
<b>then </b> <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="polyred.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T15" target="_self">Th15</a></i>
;<br/>
<a NAME="E10:36"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polyred.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E11:36"/>
<font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E5:36"><i><font color="Green">E32</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">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E2:36"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polyred.html#E10:36"><i><font color="Green">E33</font></i></a>, <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">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


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