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

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E60">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">cMGFC</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">MGFC</font> =  <a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<b>let </b><font color="Maroon" title="c4">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:72"/><i><font color="Green" title="E61">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">q</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">qc</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E3:72"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c5">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E4:72"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">qc</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">qi</font> = <font color="Maroon" title="c4">q</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E6:72"/>
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">qi</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:72"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<a NAME="E7:72"/><b>then </b><i><font color="Green" title="E64">A5</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">qi</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:72"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">qc1</font> = <font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">ir</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>
<a NAME="E11:72"/>
<font color="Maroon" title="c9">ir</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">qc2</font> = <font color="Maroon" title="c9">ir</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E14:72"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c11">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> :  <a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E15:72"/><i><font color="Green" title="E66">A7</font></i>: 
 <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="uproots.html#K10" title="UPROOTS:func.10">poly_with_roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 <b>by </b><i><a class="ref" href="uniroots.html#D5" target="_self" title="UNIROOTS:def.5">Def5</a></i>;<br/>
<a NAME="E16:72"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" title="UPROOTS:th.5">UPROOTS:5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">fs</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E18:72"/><i><font color="Green" title="E67">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" title="UPROOTS:th.5">UPROOTS:5</a></i>;<br/>
<a NAME="E19:72"/><i><font color="Green" title="E68">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c11">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#D1" title="UPROOTS:def.1">UPROOTS:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">nth</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/>
<a NAME="E21:72"/>
<font color="Maroon" title="c12">fs</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E18:72"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E22:72"/><b>then </b><i><font color="Green" title="E69">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">qc</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c14">p1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E24:72"/><i><font color="Green" title="E70">A11</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">fs</font></span>)</span> holds <br/><font color="Maroon" title="c14">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E25:72"/><i><font color="Green" title="E71">A12</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c14">p1</font> &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c14">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">qc</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b2">c</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c16">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:72_1"/><i><font color="Green" title="E72">A13</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c14">p1</font>
 <b>and </b><br/><a NAME="E2:72_1"/><i><font color="Green" title="E73">A14</font></i>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 ;<br/>

<a NAME="E3:72_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">fs</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="uniroots.html#E1:72_1"><i><font color="Green" title="E72">A13</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E4:72_1"/><b>then </b><i><font color="Green" title="E74">A15</font></i>: 
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E5:72_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">p1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:72_1"><i><font color="Green" title="E72">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:72_1"/>
<font color="Maroon" title="c14">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">qc</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c16">c</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="uniroots.html#E2:72_1"><i><font color="Green" title="E73">A14</font></i></a>, <a class="txt" href="uniroots.html#E4:72_1"><i><font color="Green" title="E74">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:72"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">p1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_2"/><i><font color="Green" title="E72">A16</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">p1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c16">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:72_2"/><i><font color="Green" title="E73">A17</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c14">p1</font>
 <b>and </b><br/><a NAME="E4:72_2"/><i><font color="Green" title="E74">A18</font></i>: 
<font color="Maroon" title="c14">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:72_2"><i><font color="Green" title="E72">A16</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:72_2"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">fs</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="uniroots.html#E3:72_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:72_2"/><b>then </b>
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:72_2"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">qc</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E25:72"><i><font color="Green" title="E71">A12</font></i></a>, <a class="txt" href="uniroots.html#E3:72_2"><i><font color="Green" title="E73">A17</font></i></a>, <a class="txt" href="uniroots.html#E4:72_2"><i><font color="Green" title="E74">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:72_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:72"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">ps</font> = <font color="Maroon" title="c14">p1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="uniroots.html#E22:72"><i><font color="Green" title="E69">A10</font></i></a>, <a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E29:72"/><i><font color="Green" title="E72">A19</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">qc</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c15">ps</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E15:72"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="uniroots.html#E19:72"><i><font color="Green" title="E68">A9</font></i></a>, <a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="uniroots.html#E25:72"><i><font color="Green" title="E71">A12</font></i></a>, <a class="ref" href="uniroots.html#T4" target="_self" title="UNIROOTS:th.4">Th4</a></i>;<br/>
<a NAME="E30:72"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c15">ps</font> holds <br/><font color="Maroon" title="c15">ps</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:72_3"/><i><font color="Green" title="E73">A20</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c15">ps</font>
 ;<br/>

<a NAME="E2:72_3"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">fs</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="uniroots.html#E1:72_3"><i><font color="Green" title="E73">A20</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:72_3"/><b>then </b>
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E4:72_3"/><b>then </b><i><font color="Green" title="E74">A21</font></i>: 
<font color="Maroon" title="c15">ps</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><font color="Maroon" title="c4">q</font>,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:72"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="uniroots.html#E25:72"><i><font color="Green" title="E71">A12</font></i></a>, <a class="txt" href="uniroots.html#E1:72_3"><i><font color="Green" title="E73">A20</font></i></a></i>;<br/>
<a NAME="E5:72_3"/><i><font color="Green" title="E75">A22</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">fs</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E24:72"><i><font color="Green" title="E70">A11</font></i></a>, <a class="txt" href="uniroots.html#E1:72_3"><i><font color="Green" title="E73">A20</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:72_3"/><b>then </b>
<font color="Maroon" title="c12">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">fs</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:72_3"/><b>then </b>
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">fs</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E5:72_3"><i><font color="Green" title="E75">A22</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span><b> such that </b><br/><a NAME="E9:72_3"/><i><font color="Green" title="E76">A23</font></i>: 
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 <b>and </b><br/><a NAME="E10:72_3"/><i><font color="Green" title="E77">A24</font></i>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E14:72"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="uniroots.html#E18:72"><i><font color="Green" title="E67">A8</font></i></a></i>;<br/>
<a NAME="E11:72_3"/>
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:72_3"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="uniroots.html#E10:72_3"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="uniroots.html#T37" target="_self" title="UNIROOTS:th.37">Th37</a></i>;<br/>
<a NAME="E12:72_3"/><b>then </b><i><font color="Green" title="E78">A25</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="uniroots.html#T26" target="_self" title="UNIROOTS:th.26">Th26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_3_1"/><i><font color="Green" title="E79">A26</font></i>: 
<font color="Maroon" title="c12">fs</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default">1,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 ;<br/><a NAME="E2:72_3_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default">1,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 <b>by </b><i><a class="ref" href="uniroots.html#T20" target="_self" title="UNIROOTS:th.20">Th20</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a></i>;<br/><b>hence </b><a NAME="E3:72_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E1:72"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="uniroots.html#E9:72_3"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="uniroots.html#E10:72_3"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="uniroots.html#E1:72_3_1"><i><font color="Green" title="E79">A26</font></i></a>, <a class="ref" href="group_1.html#T84" title="GROUP_1:th.84">GROUP_1:84</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:72_3"/>
<font color="Maroon" title="c15">ps</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2:72"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="uniroots.html#E4:72_3"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="uniroots.html#E12:72_3"><i><font color="Green" title="E78">A25</font></i></a>, <a class="ref" href="uniroots.html#T8" target="_self" title="UNIROOTS:th.8">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E31:72"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">q</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E4:72"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="uniroots.html#E7:72"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="uniroots.html#E29:72"><i><font color="Green" title="E72">A19</font></i></a>, <a class="ref" href="uniroots.html#T9" target="_self" title="UNIROOTS:th.9">Th9</a></i>;<br/>


</div>
