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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#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  non <a href="struct_0.html#V3">empty</a> <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> 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="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/><i><font color="Green">E10</font></i>:  <a href="polynom1.html#K24">-</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>
<br/>.= 
 <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="polyred.html#T3">POLYRED:3</a></i>
;<br/><a NAME="E5:10_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K22">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
 <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="txt" href="groeb_1.html#E1:10_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="groeb_1.html#E4:10_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><b>then </b> <a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T19">RLVECT_1:19</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="groeb_1.html#E1:10_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><b>hence </b><a NAME="E8:10_1_1"/>
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/>
<font color="Maroon">c<sub>4</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>
 ;<br/><div class="add"><a NAME="E2:10_1_2"/><b>then </b><i><font color="Green">E9</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></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/><a NAME="E3:10_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
(  <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>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E4:10_1_2"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></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="groeb_1.html#E2:10_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="groeb_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:10_1_2"/><b>then </b>
(  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</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"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E6:10_1_2"/><b>then </b>
(  <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> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</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> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:10_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="groeb_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:10_1_2"/><b>then </b>
(  <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="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><b>hence </b><a NAME="E8:10_1_2"/>
 <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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