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<b>let </b><font color="Maroon">c<sub>1</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="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:57_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:57"><i><font color="Green">E44</font></i></a>, <a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/><a NAME="E3:57_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a></i>;<br/><b>hence </b><a NAME="E4:57_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E2:57"><i><font color="Green">E45</font></i></a>, <a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:57"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is <a href="uproots.html#V1" target="_self">non-zero</a>
 
<b>by </b><i><a class="ref" href="uproots.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:57"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="uproots.html#E1:57"><i><font color="Green">E44</font></i></a>, <a class="ref" href="uproots.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E6:57"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E7:57"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="uproots.html#E3:57"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:57"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E6:57"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E10:57"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E6:57"><i><font color="Green">E47</font></i></a>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E9:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="uproots.html#T39" target="_self">Th39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E10:57"><i><font color="Green">E49</font></i></a>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E12:57"/>
<font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:57"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uproots.html#E2:57"><i><font color="Green">E45</font></i></a>, <a class="txt" href="uproots.html#E6:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="uproots.html#E9:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>


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