<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> : (  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green">E59</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( ( (  not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E4:66"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:66"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:66"/><i><font color="Green">E62</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E3:66_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E8:66"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:66"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="uniroots.html#E6:66"><i><font color="Green">E63</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> ) </span>}</span> ;<br/>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:66_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>)
 ;<br/>

<a NAME="E2:66_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:66_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:66_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> );<br/>
<a NAME="E11:66"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">H<sub>1</sub></font>(1) as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="txt" href="uniroots.html#E10:66"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E2:66_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(1)
 ;<br/>

<a NAME="E3:66_3"/><i><font color="Green">E67</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:66"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E5:66"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:66_3"/><i><font color="Green">E68</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:66"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E5:66"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E6:66_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:66"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E5:66"><i><font color="Green">E62</font></i></a>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:66_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:66_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E9:66_3"/><i><font color="Green">E70</font></i>: 
1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_3_1"/>
not <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E5:66_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:66_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:66_3_1"/><i><font color="Green">E73</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_3_1"><i><font color="Green">E71</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E9:66_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E10:66_3_1"/><i><font color="Green">E75</font></i>: 
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E6:66_3_1"><i><font color="Green">E72</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_3_1_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:66_3_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E10:66_3_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>thus </b><a NAME="E4:66_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_3_1_1"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E12:66_3_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E13:66_3_1"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E14:66_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E9:66_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="uniroots.html#E7:66_3_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E9:66_3_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uniroots.html#E10:66_3_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:66_3"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="uproots.html#T11">UPROOTS:11</a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
 <a href="group_4.html#K3">Product</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E9:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uniroots.html#E3:66_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="uniroots.html#E6:66_3"><i><font color="Green">E69</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E3:66"><i><font color="Green">E60</font></i></a>, <a class="txt" href="uniroots.html#E4:66_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="uniroots.html#E9:66_3"><i><font color="Green">E70</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:66_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E11:66_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="uproots.html#T62">UPROOTS:62</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:66"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66_4"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E2:66_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>

<a NAME="E3:66_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:66_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:66_4"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:66_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:66_4"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:66_4"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E8:66_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uniroots.html#E3:66_4"><i><font color="Green">E70</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E10:66_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uniroots.html#E5:66_4"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E11:66_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E12:66_4"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E13:66_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:66_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/>
<a NAME="E14:66_4"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T24">FINSEQ_2:24</a></i>;<br/>
<a NAME="E15:66_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E16:66_4"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="txt" href="uniroots.html#E6:66_4"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E19:66_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E12:66_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="uniroots.html#E14:66_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E20:66_4"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="group_4.html#T9">GROUP_4:9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E22:66_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>)
 <b>and </b><br/><a NAME="E23:66_4"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:66_4"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<a NAME="E24:66_4"/><i><font color="Green">E82</font></i>: 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E8:66_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uniroots.html#E20:66_4"><i><font color="Green">E79</font></i></a>, <a class="txt" href="uniroots.html#E23:66_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1) as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="txt" href="uniroots.html#E10:66"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E26:66_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1)
 ;<br/>

<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_4_1"/>
(  not <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or ( <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_4_1_1"><b>suppose </b></a><a NAME="E1:66_4_1_1"/><i><font color="Green">E83</font></i>: 
(  not <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_4_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_4_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E4:66_4_1_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:66_4_1_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E6:66_4_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_1_1_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E7:66_4_1_1_1_1"/><b>then </b>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66_4_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uniroots.html#E5:66_4_1_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E8:66_4_1_1_1_1"/><b>then </b>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:66_4_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:66_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="uniroots.html#E3:66_4_1_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_1_1_1"><i><font color="Green">E86</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:66_4_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E4:66_4_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E5:66_4_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:66_4_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:66_4"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E7:66_4_1_1_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E6:66_4_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E8:66_4_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:66_4_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="uniroots.html#E5:66_4_1_1_1"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:66_4_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66"><i><font color="Green">E60</font></i></a>, <a class="txt" href="uniroots.html#E6:66_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E1:66_4_1_1"><i><font color="Green">E83</font></i></a></i>;<br/><b>then </b><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
 <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E10:66_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="uniroots.html#E16:66_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="uniroots.html#E24:66_4"><i><font color="Green">E82</font></i></a>, <a class="ref" href="uproots.html#T34">UPROOTS:34</a></i>
<br/>.= 
 <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E2:66_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>
;<br/><b>hence </b><a NAME="E5:66_4_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_4_1_2"><b>suppose </b></a><a NAME="E1:66_4_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>18</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:66_4_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> :  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 </span>}</span> 
 <b>and </b><br/><a NAME="E4:66_4_1_2"/><i><font color="Green">E85</font></i>: 
 <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="funct_2.html#R2">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i><a class="ref" href="uniroots.html#D5" target="_self">Def5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font>,1 <a href="uproots.html#K2">-bag</a> ;<br/><a NAME="E5:66_4_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66"><i><font color="Green">E60</font></i></a>, <a class="txt" href="uniroots.html#E6:66_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E1:66_4_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_2"><i><font color="Green">E85</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_4_1_2_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_4_1_2_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:66_4_1_2_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:66_4_1_2_1_1"/><i><font color="Green">E89</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E6:66_4_1_2_1_1"/>
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> or  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="uniroots.html#E5:66_4_1_2_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E7:66_4_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E22:66_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="uniroots.html#E3:66_4_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_2_1_1"><i><font color="Green">E88</font></i></a></i>;<br/><b>hence </b><a NAME="E8:66_4_1_2_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_4_1_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:66_4_1_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_4_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_4_1_2_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E2:66_4_1_2_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_4_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:66_4_1_2_1_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_4_1_2_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:66_4_1_2_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:66_4_1_2_1_2_1"/><i><font color="Green">E90</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:66_4"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E6:66_4_1_2_1_2_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E5:66_4_1_2_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E7:66_4_1_2_1_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_4_1_2_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_2_1_2_1"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_4_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:66_4_1_2_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_4_1_2_1_2_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:66_4_1_2_1_2_2"/><i><font color="Green">E89</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_4_1_2"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E5:66_4_1_2_1_2_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniroots.html#E1:66_4_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_2_1_2_2"><i><font color="Green">E89</font></i></a></i>;<br/><b>hence </b><a NAME="E6:66_4_1_2_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_4_1_2_1_2_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_2_1_2_2"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:66_4_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E8:66_4_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_4_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_4_1_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:66_4_1_2_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:66_4_1_2_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E6:66_4_1_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E7:66_4_1_2_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:66_4_1_2_2"/><i><font color="Green">E92</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E22:66_4"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E10:66_4_1_2_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E11:66_4_1_2_2"/><i><font color="Green">E94</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E3:66_4_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="uniroots.html#E4:66_4_1_2_2"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>thus </b><a NAME="E12:66_4_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E6:66_4_1_2_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="uniroots.html#E8:66_4_1_2_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="uniroots.html#E10:66_4_1_2_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="uniroots.html#E11:66_4_1_2_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E9:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uniroots.html#E5:66_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E16:66_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="uniroots.html#E24:66_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="uniroots.html#E5:66_4_1_2"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
 <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T66">UPROOTS:66</a></i>
<br/>.= 
 <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E7:66_4_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="uniroots.html#E8:66_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="uproots.html#T12">UPROOTS:12</a></i>
;<br/><b>hence </b><a NAME="E10:66_4_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:66"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4">POLYNOM2:sch 4</a>(<a class="txt" href="uniroots.html#E11:66"><i><font color="Green">E66</font></i></a>, <a class="txt" href="uniroots.html#E12:66"><i><font color="Green">E67</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="txt" href="uniroots.html#E10:66"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_5_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E3:66_5_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E4:66_5_1"/><i><font color="Green">E71</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:66_5_1"/>
( not  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E6:66_5_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:66_5_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/><b>hence </b><a NAME="E7:66_5_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:66_5_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="uniroots.html#E3:66_5_1"><i><font color="Green">E70</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:66_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E3:66_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E4:66_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E16:66"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E17:66"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E19:66"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>)
 <b>and </b><br/><a NAME="E20:66"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:66"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E5:66"><i><font color="Green">E62</font></i></a>, <a class="txt" href="uniroots.html#E13:66"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E21:66"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:66"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E22:66"/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:66"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E9:66"><i><font color="Green">E64</font></i></a>, <a class="txt" href="uniroots.html#E16:66"><i><font color="Green">E69</font></i></a>, <a class="txt" href="uniroots.html#E17:66"><i><font color="Green">E70</font></i></a>, <a class="txt" href="uniroots.html#E19:66"><i><font color="Green">E71</font></i></a>, <a class="txt" href="uniroots.html#E20:66"><i><font color="Green">E72</font></i></a></i>;<br/>


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