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<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">f</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">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:68"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</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> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">i</font> )
 ;<br/>

<a NAME="E4:68"/><i><font color="Green">E40</font></i>: 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="polynom4.html#T21">POLYNOM4:21</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="polynom3.html#K13">1_.</a> <a href="complfld.html#K1">F_Complex</a>  = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T28">POLYNOM5:28</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="ref" href="polynom5.html#T30">POLYNOM5:30</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1"><b>suppose </b></a><a NAME="E1:68_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:68_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> 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#T32">FINSEQ_1:32</a></i>;<br/><b>then </b><font color="Maroon">p</font> = 
 <a href="group_1.html#K2">1.</a> <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="group_4.html#T11">GROUP_4:11</a></i>
<br/>.= 
 <a href="polynom3.html#K13">1_.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="polynom3.html#T37">POLYNOM3:37</a></i>
<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <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="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/><b>hence </b><a NAME="E4:68_2_1"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2"><b>suppose </b></a><a NAME="E1:68_2_2"/><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:68_2_2"/><b>then </b><i><font color="Green">E53</font></i>: 
0 <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">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">fi</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">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">fi</font>;<br/><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:68_2_2_1"/>
<font color="Maroon">i</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">f</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">fi</font> = <font color="Maroon">f</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">i</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">fi</font> = <font color="Maroon">f</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">i</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">x</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">fi</font>;<br/><b>take </b><font color="Maroon">x</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">fi</font>;<br/><b>thus </b><a NAME="E3:68_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">x</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">F</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="E5:68_2_2"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</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">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:68_2_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">f</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="uniroots.html#E10:68_2_2"><i><font color="Green">E67</font></i></a>);<br/></i><a NAME="E7:68_2_2"/><i><font color="Green">E56</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">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:68_2_2"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">r</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="polynom4.html#K2">eval</a> <font color="Olive">r</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a> );<br/><a NAME="E9:68_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</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="E2:68_2_2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">f1</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">f</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#E3:68_2_2_2"><i><font color="Green">E66</font></i></a>, </i>;<br/>
<a NAME="E3:68_2_2_2"/><i><font color="Green">E66</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fd1</font> = <font color="Maroon">f</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/>
<b>reconsider </b><font color="Maroon">fd1</font> = <font color="Maroon">fd1</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/>
<b>take </b>
<font color="Maroon">fd1</font>
;<br/>

<b>thus </b><font color="Maroon">fd1</font> = 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, </i>
;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_2_2_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <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> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E2:68_2_2_2"><i><font color="Green">E62</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_2_2_1"><b>suppose </b></a><a NAME="E1:68_2_2_2_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_2_2_2_1"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">fd1</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="polynom4.html#T21">POLYNOM4:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_2_2_2"><b>suppose </b></a><a NAME="E1:68_2_2_2_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_2_2_2_2"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">fd1</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E9:68_2_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="uniroots.html#E7:68_2_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:68_2_2_3"/><i><font color="Green">E68</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:68_2_2_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><b>assume </b><a NAME="E3:68_2_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E5:68_2_2_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E6:68_2_2_3"/><i><font color="Green">E71</font></i>: 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">r</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 ;<br/><a NAME="E7:68_2_2_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</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">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E8:68_2_2_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E9:68_2_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:68_2_2_3"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</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">f</font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E8:68_2_2_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E9:68_2_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:68_2_2_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</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">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:68_2_2_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</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">f</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">fi</font> = <font color="Maroon">f</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">i</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>reconsider </b><font color="Maroon">fi1</font> = <font color="Maroon">f</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">i</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="E13:68_2_2_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</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="E14:68_2_2_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi1</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">i</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:68_2_2_3"/>
<font color="Maroon">i</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">i</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">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E17:68_2_2_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><a NAME="E16:68_2_2_3"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</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="E17:68_2_2_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</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">i</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="E18:68_2_2_3"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</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">i</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">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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">fi1d1</font> = <font color="Maroon">fi1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</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="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">fi1d1p</font> = <font color="Maroon">fi1d1</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="E21:68_2_2_3"/>
<font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">fi1d1</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/><a NAME="E22:68_2_2_3"/><b>then </b><i><font color="Green">E170</font></i>: 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">fi1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">fi</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">fi1d1</font>
 <b>by </b><i><a class="ref" href="group_4.html#T9">GROUP_4:9</a></i>;<br/><b>reconsider </b><font color="Maroon">pfi1</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">fi1</font>, <font color="Maroon">pfi</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">fi</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/><b>thus </b><a NAME="E24:68_2_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">pfi1</font>
;<br/>

<b>thus </b><a NAME="E1:68_2_2_3_1"/>
<font color="Maroon">pfi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:68_2_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="uniroots.html#T4" target="_self">Th4</a></i>;<br/>

<b>reconsider </b><font color="Maroon">epfi</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">pfi</font>,<font color="Maroon">c</font>, <font color="Maroon">efi1d1p</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">fi1d1p</font>,<font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">iepfi</font> = <font color="Maroon">epfi</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:68_2_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="uniroots.html#E10:68_2_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="uniroots.html#E14:68_2_2_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="uniroots.html#E16:68_2_2_3"><i><font color="Green">E77</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_3_1_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_2_3_1_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</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> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">i1</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_2_3_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_2_3_1_1_1_1"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">fi1d1p</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_2_3_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_2_3_1_1_1_2"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">fi1d1p</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="uniroots.html#E9:68_2_2"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="uniroots.html#E7:68_2_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">iefi1d1p</font> = <font color="Maroon">efi1d1p</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E6:68_2_2_3_1"/>
<font color="Maroon">pfi1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">pfi</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">fi1d1p</font>
 
<b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>then </b> <a href="polynom4.html#K2">eval</a> <font color="Maroon">pfi1</font>,<font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">pfi</font>,<font color="Maroon">c</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">fi1d1p</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>
<br/>.= 
<font color="Maroon">iepfi</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">iefi1d1p</font>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
;<br/>
<b>hence </b><a NAME="E8:68_2_2_3_1"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">pfi1</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E11:68_2_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S4">POLYNOM2:sch 4</a>(, );<br/></i><b>then consider </b><font color="Maroon">r</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E13:68_2_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</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">f</font></span>)</span>
 <b>and </b><br/><a NAME="E14:68_2_2"/><i><font color="Green">E187</font></i>: 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">r</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i/>;<br/><a NAME="E15:68_2_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T55">FINSEQ_3:55</a></i>;<br/><b>hence </b><a NAME="E16:68_2_2"/>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">c</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
