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

<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> , <font color="Maroon">ni</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">q</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:70"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">ni</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">ni</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<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><a NAME="E2:70"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">qc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<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">l</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:70"/><i><font color="Green">E39</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="E4:70"/><i><font color="Green">E40</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>
 <b>and </b><br/><a NAME="E5:70"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">l</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">ni</font></span>)</span>,<font color="Maroon">qc</font>
 ;<br/>

<b>set </b><font color="Maroon">ttt</font> = <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">ni</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font></span>)</span>;<br/>
<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>, <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="E7:70"/><i><font color="Green">E42</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="E8:70"/><i><font color="Green">E53</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="E9:70"/><i><font color="Green">E54</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> <font color="Maroon">ni</font> 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> <font color="Maroon">ni</font> &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="E10:70"/><i><font color="Green">E55</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> ,<font color="Maroon">ni</font></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><a class="txt" href="uniroots.html#E18:70"><i><font color="Green">E62</font></i></a>, </i>;<br/>
<a NAME="E11:70"/><i><font color="Green">E56</font></i>: 
 <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> <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> ,<font color="Maroon">ni</font></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_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="E1:70_1"/><i><font color="Green">E58</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="70_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_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> <font color="Maroon">ni</font> 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> <font color="Maroon">ni</font> &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="70_1_1_1"><b>suppose </b></a><a NAME="E1:70_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> <font color="Maroon">ni</font> 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="E2:70_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="70_1_1_2"><b>suppose </b></a><a NAME="E1:70_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> <font color="Maroon">ni</font> &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="E2:70_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>
<a NAME="E13:70"/><b>then </b>
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">qc</font> is <a href="int_1.html#V1">integer</a>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E15:70"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">qc</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">jc</font> = <font color="Maroon">j</font>, <font color="Maroon">lc</font> = <font color="Maroon">l</font>, <font color="Maroon">mc</font> = <font color="Maroon">m</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="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">jcf</font> = <font color="Maroon">jc</font>, <font color="Maroon">lcf</font> = <font color="Maroon">lc</font>, <font color="Maroon">mcf</font> = <font color="Maroon">mc</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E18:70"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">lcf</font> <a href="group_1.html#K10">*</a> <font color="Maroon">jcf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</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/>
<i><font color="Green">E66</font></i>: <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">lcf</font> <a href="group_1.html#K10">*</a> <font color="Maroon">jcf</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">mcf</font>
<b>by </b><i>, , <a class="txt" href="uniroots.html#E19:70"><i><font color="Green">E66</font></i></a>, , , <a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_3_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_3_1_1"><b>suppose </b></a><a NAME="E1:70_3_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:70_3_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:70_3_1_1"/><b>then </b>
<font color="Maroon">l</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/><a NAME="E4:70_3_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">l</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, <a class="ref" href="wsierp_1.html#T22">WSIERP_1:22</a></i>;<br/><a NAME="E5:70_3_1_1"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">l</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/><a NAME="E6:70_3_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="int_1.html#K5">div</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>;<br/><a NAME="E7:70_3_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="int_1.html#K5">div</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E8:70_3_1_1"/>
<font color="Maroon">j</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font> <a href="int_1.html#K5">div</a> <font color="Maroon">l</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_3_1_2"><b>suppose </b></a><a NAME="E1:70_3_1_2"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:70_3_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="int_1.html#K5">div</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#T75">INT_1:75</a></i>;<br/><b>hence </b><a NAME="E3:70_3_1_2"/>
<font color="Maroon">j</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font> <a href="int_1.html#K5">div</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="int_2.html#T16">INT_2:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E21:70"/>
<font color="Maroon">j</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font> <a href="int_1.html#K5">div</a> <font color="Maroon">l</font>
 ;<br/>


</div>
