<?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>assume </b><b>that </b><br/><a NAME="E1:67"/><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>
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">ni</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">cPRFC</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">cMGFC</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">d</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 <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> &amp; ( (  not <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> or <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> or <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) implies <font color="Maroon">a<sub>2</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">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp; <font color="Olive">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">d</font> ) );<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_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="E1:67_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:67_1"/><b>then </b><i><font color="Green">E41</font></i>: 
not <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="binarith.html#T5">BINARITH:5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_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="67_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1"/><i><font color="Green">E42</font></i>: 
(  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"><a NAME="E2:67_1_1_1"/>
<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> 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="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="E3:67_1_1_1"/>
 ex <font color="Olive">x</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="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">x</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_2"/><i><font color="Green">E53</font></i>: 
( <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>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> 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>  <b>by </b><i><a class="txt" href="uniroots.html#E10:67"><i><font color="Green">E62</font></i></a>, <a class="ref" href="binarith.html#T5">BINARITH:5</a></i>;<br/><a NAME="E3:67_1_1_2"/>
 <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">i'</font> 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="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="E4:67_1_1_2"/>
 ex <font color="Olive">x</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="polynom3.html#K16">Polynom-Ring</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">x</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></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="E5:67"/><i><font color="Green">E54</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="E6:67"/><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> <font color="Maroon">n</font> 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#E9:67"><i><font color="Green">E61</font></i></a>);<br/></i>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><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="E1:67_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:67_2"/><b>then </b>
 ex <font color="Olive">d</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 <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> &amp; ( (  not <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> or <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> or <font color="Olive">d</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="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> ) &amp; ( <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp; <font color="Olive">d</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="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Olive">d</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:67_2"/>
( ( (  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> ) implies <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> ) &amp; ( <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> implies <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> ) )
 ;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>set </b><font color="Maroon">rbp</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> 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">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> ) </span>}</span> ;<br/>
<a NAME="E8:67"/><i><font color="Green">E58</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> 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">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:67_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> 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">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:67_3"/><b>then </b>
 ex <font color="Olive">y</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">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 
;<br/>
<b>hence </b><a NAME="E3:67_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:67"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<b>by </b><i>, <a class="txt" href="uniroots.html#E1:67"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E10:67"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="tarski.html#R1">c=</a> 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>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">rbp</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> 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">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> ) </span>}</span>  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="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">bp</font> = <font color="Maroon">rbp</font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<a NAME="E12:67"/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">f</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">rbp</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E10:67"><i><font color="Green">E62</font></i></a>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E14:67"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font>
 ;<br/>

<b>thus </b><a NAME="E15:67"/>
 <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>by </b><i><a class="txt" href="uniroots.html#E10:67"><i><font color="Green">E62</font></i></a></i>;<br/>

<b>thus </b><a NAME="E16:67"/>
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>by </b><i/>;<br/>

<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<b>set </b><font color="Maroon">bi</font> = <span class="p1">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<b>consider </b><font color="Maroon">rbn</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="E18:67"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">rbn</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> 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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E19:67"/><i><font color="Green">E67</font></i>: 
 <a href="uniroots.html#K6" target="_self">cyclotomic_poly</a> <font color="Maroon">n</font> <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">rbn</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">bn</font> = <font color="Maroon">rbn</font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<b>set </b><font color="Maroon">rbibn</font> = <span class="p1">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbn</font>;<br/>
<b>reconsider </b><font color="Maroon">rbibn</font> = <span class="p1">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbn</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>  ;<br/>
<b>set </b><font color="Maroon">bibn</font> = <font color="Maroon">rbibn</font>,1 <a href="uproots.html#K2">-bag</a> ;<br/>
<a NAME="E21:67"/>
<font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="subset_1.html#R1">misses</a> <font color="Maroon">rbn</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_4"/>
<span class="p1">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">rbn</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">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:67_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">rbn</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:67_4"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbn</font> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E6:67_4"><i><font color="Green">E70</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">y</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="E6:67_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E7:67_4"/><i><font color="Green">E71</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:67"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">y1</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="E9:67_4"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E10:67_4"/><i><font color="Green">E73</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:67_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="uniroots.html#E7:67_4"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:67_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="uniroots.html#E9:67_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="uniroots.html#E10:67_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="uniroots.html#E22:67"><i><font color="Green">E74</font></i></a>, <a class="txt" href="uniroots.html#E23:67"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:67"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">rbn</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbn</font></span>)</span>,1 <a href="uproots.html#K2">-bag</a> 
 
<b>by </b><i><a class="ref" href="uproots.html#T12">UPROOTS:12</a></i>;<br/>
<a NAME="E23:67"/><i><font color="Green">E75</font></i>: 
 <a href="uniroots.html#K4" target="_self">unital_poly</a> <a href="complfld.html#K1">F_Complex</a> ,<font color="Maroon">ni</font> <a href="funct_2.html#R2">=</a>  <a href="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:67_5_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> 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="uniroots.html#K1" target="_self">MultGroup</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <b>by </b><i><a class="txt" href="uniroots.html#E19:67"><i><font color="Green">E67</font></i></a></i>;<br/><a NAME="E3:67_5_1"/><i><font color="Green">E77</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_5_1_1"/>
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or (  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> ) or (  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_1_1_1"><b>suppose </b></a><a NAME="E1:67_5_1_1_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:67_5_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbn</font>
 <b>by </b><i/>;<br/><a NAME="E3:67_5_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:67_5_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_1_1_2"><b>suppose </b></a><a NAME="E1:67_5_1_1_2"/>
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> )
 ;<br/><div class="add"><a NAME="E2:67_5_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbp</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:67_5_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_1_1_3"><b>suppose </b></a><a NAME="E1:67_5_1_1_3"/>
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> )
 ;<br/><div class="add"><a NAME="E2:67_5_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i/>;<br/><a NAME="E3:67_5_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:67_5_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:67_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbp</font>
 ;<br/><a NAME="E3:67_5"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbp</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_5_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbn</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbp</font> )
 </span><b>by </b><i>, <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="67_5_2_1"><b>suppose </b></a><a NAME="E1:67_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E18:67"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_2_2"><b>suppose </b></a><a NAME="E1:67_5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbn</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</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="E3:67_5_2_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:67_5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_5_2_3"><b>suppose </b></a><a NAME="E1:67_5_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbp</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</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="E3:67_5_2_3"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 ;<br/><b>thus </b><a NAME="E4:67_5_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:67"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">rbibn</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">rbp</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E26:67"/>
<font color="Maroon">rbibn</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">rbp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_6"/>
<font color="Maroon">rbibn</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">rbp</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">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:67_6"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">rbp</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:67_6"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbibn</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbp</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</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="E6:67_6"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E7:67_6"/><i><font color="Green">E206</font></i>: 
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_6_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbn</font> )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E22:67"><i><font color="Green">E74</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="67_6_1_1"><b>suppose </b></a><a NAME="E1:67_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ni</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 ;<br/><div class="add"><a NAME="E2:67_6_1_1"/><b>then </b>
 ex <font color="Olive">y</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">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">ni</font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E2:67"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E3:67_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E23:67"><i><font color="Green">E75</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_6_1_2"><b>suppose </b></a><a NAME="E1:67_6_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rbn</font>
 ;<br/><div class="add"><a NAME="E2:67_6_1_2"/><b>then </b>
 ex <font color="Olive">y</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">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp;  <a href="group_1.html#K7">ord</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:67_6_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E23:67"><i><font color="Green">E75</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E27:67"/><b>then </b><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">rbibn</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">rbp</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="uproots.html#T12">UPROOTS:12</a></i>;<br/>
 <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="uproots.html#K10">poly_with_roots</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon">rbibn</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">rbp</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/>.= 
<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> <span class="p1">(<span class="default"><a href="uproots.html#K10">poly_with_roots</a> <span class="p2">(<span class="default"><font color="Maroon">rbp</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/>
<b>hence </b><a NAME="E29:67"/>
 <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#E10:67"><i><font color="Green">E62</font></i></a>, , </i>;<br/>


</div>
