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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  <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="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-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>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</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/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</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/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <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> <b>by </b><i><a class="txt" href="groeb_1.html#E4:39_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E6:39_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:39"><i><font color="Green">E30</font></i></a>, <a class="txt" href="groeb_1.html#E4:39_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E8:39_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="polyred.html#R9">is_top_reducible_wrt</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D12">POLYRED:def 12</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:39_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="polyred.html#R8">top_reduces_to</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E8:39_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="polyred.html#D11">POLYRED:def 11</a></i>;<br/><a NAME="E11:39_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="polyred.html#R3">reduces_to</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>, <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:39_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="polyred.html#D10">POLYRED:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:39_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K15">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="vectsp_1.html#K5">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="polyred.html#K1">*'</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><a NAME="E14:39_1"/>
<font color="Maroon">c<sub>9</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="txt" href="groeb_1.html#E11:39_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polyred.html#D5">POLYRED:def 5</a></i>;<br/><a NAME="E15:39_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E8:39_1"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E16:39_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E17:39_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:39_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="groeb_1.html#E13:39_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom1.html#T54">POLYNOM1:54</a></i>;<br/><b>hence </b><a NAME="E18:39_1"/>
ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E16:39_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
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="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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