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<b>let </b><font color="Maroon">n</font> be  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/><b>let </b><font color="Maroon">j</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">k</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">q</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">qc</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">qc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">qc</font>
 <b>and </b><br/><a NAME="E3:69"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">qc</font>
 ;<br/>

<b>set </b><font color="Maroon">jfc</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">qc</font>;<br/>
<b>reconsider </b><font color="Maroon">jc</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">qc</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="uproots.html#T1">UPROOTS:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>suppose </b></a><a NAME="E1:69_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_1"/>
<font color="Maroon">j</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2"><b>suppose </b></a><a NAME="E1:69_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2"/>
1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/><b>then 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>, <font color="Maroon">p</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="E4:69_1_2"/><i><font color="Green">E41</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="E5:69_1_2"/><i><font color="Green">E42</font></i>: 
 <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> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E6:69_1_2"/><i><font color="Green">E53</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>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>( ( (  not <font color="Olive">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> or <font color="Olive">i</font> <a href="nat_1.html#R1">divides</a> 1 or <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) implies <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> ) &amp; ( <font color="Olive">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Olive">i</font> <a href="nat_1.html#R1">divides</a> 1 &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> implies <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> ) )
 <b>and </b><br/><a NAME="E7:69_1_2"/><i><font color="Green">E54</font></i>: 
 <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">n</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">epfc</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">qc</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be  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/><b>assume </b><a NAME="E1:69_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_2_1_1"/>
(  not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or ( <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_1"/>
(  not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_1_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_2"/>
( <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_1_1_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">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="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">ep</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">qc</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">epc</font> =  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">qc</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>set </b><font color="Maroon">eup1fc</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span>,<font color="Maroon">qc</font>;<br/><b>reconsider </b><font color="Maroon">eup1</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span>,<font color="Maroon">qc</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">eup1c</font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span>,<font color="Maroon">qc</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/><a NAME="E13:69_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p3">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">qc</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">p</font>,<font color="Maroon">qc</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>;<br/><a NAME="E14:69_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span>,<font color="Maroon">qc</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">qc</font></span>)</span></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">p</font>,<font color="Maroon">qc</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>;<br/><a NAME="E15:69_1_2"/>
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,1</span>)</span>,<font color="Maroon">qc</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">qc</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">eup1c</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">jc</font>
 <b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E16:69_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">eup1c</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">jc</font></span>)</span> <a href="binop_2.html#K5">*</a> <font color="Maroon">epc</font>
 <b>by </b><i>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>;<br/><a NAME="E17:69_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">eup1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">ep</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E18:69_1_2"/>
<font color="Maroon">j</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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