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

<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="vectsp_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a>;<br/>

<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E44">A1</font></i>: 
not <font color="Maroon" title="c1">R</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">Z</font> =  <a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">cZ</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">q</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">vR</font> =  <a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c6">n</font> =  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">Rs</font> =  <a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c8">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c9">cRs</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> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">cZs</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>;<br/>
<a NAME="E2:52"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T32" target="_self" title="WEDDWITT:th.32">Th32</a></i>;<br/>
<a NAME="E3:52"/><b>then </b><i><font color="Green" title="E46">A3</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="uniroots.html#T21" title="UNIROOTS:th.21">UNIROOTS:21</a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green" title="E47">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T21" target="_self" title="WEDDWITT:th.21">Th21</a></i>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">natq1</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E7:52"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T33" target="_self" title="WEDDWITT:th.33">Th33</a></i>;<br/>
<a NAME="E8:52"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:52"/><b>then </b><i><font color="Green" title="E49">A6</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E50">A7</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:52_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Z</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:52"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>;<br/><a NAME="E3:52_1"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:52_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="prepower.html#T45" title="PREPOWER:th.45">PREPOWER:45</a></i>;<br/><b>hence </b><a NAME="E4:52_1"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:52"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="weddwitt.html#T22" target="_self" title="WEDDWITT:th.22">Th22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:52"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E9:52"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c13">B</font> = <span class="p1">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> ;<br/>
<a NAME="E12:52"/>
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> <span class="p1">{<span class="default"> <font color="Olive" title="b2">X</font> where <font color="Olive" title="b2">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 ;<br/>

<a NAME="E2:52_2"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
;<br/>
<b>hence </b><a NAME="E3:52_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:52"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E14:52"/><b>then </b><i><font color="Green" title="E52">A10</font></i>: 
 <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p3">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E16:52"/><i><font color="Green" title="E53">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E17:52"/><i><font color="Green" title="E54">A12</font></i>: 
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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E18:52"/><i><font color="Green" title="E55">A13</font></i>: 
<font color="Maroon" title="c14">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:52_3"/>
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f</font> &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> )
 ;<br/>

<a NAME="E2:52_3"/><b>then </b>
( <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E17:52"><i><font color="Green" title="E54">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:52_3"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>now </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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:52_4_1"/>
<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">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c16">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_4_1"/><i><font color="Green" title="E56">A14</font></i>: 
<font color="Maroon" title="c16">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f</font>
 <b>and </b><br/><a NAME="E4:52_4_1"/><i><font color="Green" title="E57">A15</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">xx</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:52_4_1"/><i><font color="Green" title="E58">A16</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">xx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E17:52"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_4_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="weddwitt.html#E4:52_4_1"><i><font color="Green" title="E57">A15</font></i></a></i>;<br/><a NAME="E6:52_4_1"/><i><font color="Green" title="E59">A17</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">X</font> = <font color="Maroon" title="c15">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_4_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="weddwitt.html#E5:52_4_1"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">xx</font> = <font color="Maroon" title="c16">xx</font> as   <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> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_4_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="weddwitt.html#E6:52_4_1"><i><font color="Green" title="E59">A17</font></i></a></i>;<br/><a NAME="E9:52_4_1"/>
<font color="Maroon" title="c18">xx</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_4_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E10:52_4_1"/><b>then </b>
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c18">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">xx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="group_5.html#T93" title="GROUP_5:th.93">GROUP_5:93</a></i>;<br/><a NAME="E11:52_4_1"/><b>then </b><i><font color="Green" title="E60">A18</font></i>: 
<font color="Maroon" title="c17">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:52_4_1"><i><font color="Green" title="E58">A16</font></i></a></i>;<br/><a NAME="E12:52_4_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:52_4_1"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/><b>hence </b><a NAME="E13:52_4_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E11:52_4_1"><i><font color="Green" title="E60">A18</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:52_4"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c16">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E4:52_4"/><i><font color="Green" title="E56">A19</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">X</font>
 <b>and </b><br/><a NAME="E5:52_4"/><i><font color="Green" title="E57">A20</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>and </b><br/><a NAME="E6:52_4"/><i><font color="Green" title="E58">A21</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><b>consider </b><font color="Maroon" title="c17">a</font> being    <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> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E8:52_4"/><i><font color="Green" title="E59">A22</font></i>: 
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">X</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:52_4"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="weddwitt.html#T14" target="_self" title="WEDDWITT:th.14">Th14</a></i>;<br/><a NAME="E9:52_4"/><i><font color="Green" title="E60">A23</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c17">a</font>
 <b>by </b><i><a class="ref" href="group_3.html#T98" title="GROUP_3:th.98">GROUP_3:98</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:52_4"/><i><font color="Green" title="E61">A24</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">xx</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:52_4"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E12:52_4"/><i><font color="Green" title="E62">A25</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">xx</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:52_4"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="weddwitt.html#E9:52_4"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="weddwitt.html#E11:52_4"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:52_4"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:52_4"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="weddwitt.html#E11:52_4"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="group_5.html#T93" title="GROUP_5:th.93">GROUP_5:93</a></i>;<br/><a NAME="E14:52_4"/><b>then </b><i><font color="Green" title="E63">A26</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E15:52_4"/><b>then </b>
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E17:52"><i><font color="Green" title="E54">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E16:52_4"/>
<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">f</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E4:52_4"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="weddwitt.html#E11:52_4"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="weddwitt.html#E12:52_4"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="weddwitt.html#E14:52_4"><i><font color="Green" title="E63">A26</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:52"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E21:52"/><b>then </b><i><font color="Green" title="E56">A27</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E16:52"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="weddwitt.html#E18:52"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E22:52"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">natq1</font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T38" target="_self" title="WEDDWITT:th.38">Th38</a></i>;<br/>
<a NAME="E23:52"/><b>then </b><i><font color="Green" title="E57">A28</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">natq1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E21:52"><i><font color="Green" title="E56">A27</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<a NAME="E24:52"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E13:52"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">f1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E26:52"/><i><font color="Green" title="E58">A29</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 <b>and </b><br/><a NAME="E27:52"/><i><font color="Green" title="E59">A30</font></i>: 
<font color="Maroon" title="c15">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">f2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E29:52"/><i><font color="Green" title="E60">A31</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 <b>and </b><br/><a NAME="E30:52"/><i><font color="Green" title="E61">A32</font></i>: 
<font color="Maroon" title="c16">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font>;<br/>
<a NAME="E31:52"/><i><font color="Green" title="E62">A33</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E14:52"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="weddwitt.html#E26:52"><i><font color="Green" title="E58">A29</font></i></a>, <a class="txt" href="weddwitt.html#E29:52"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:52_5"/><i><font color="Green" title="E63">A34</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p3">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> </span>)</span>
 ;<br/><a NAME="E3:52_5"/>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:52_5"><i><font color="Green" title="E63">A34</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:52_5"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E33:52"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p3">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E34:52"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">f2</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E26:52"><i><font color="Green" title="E58">A29</font></i></a>, <a class="txt" href="weddwitt.html#E29:52"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E35:52"/><b>then </b><i><font color="Green" title="E63">A35</font></i>: 
<font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:52"><i><font color="Green" title="E59">A30</font></i></a>, <a class="txt" href="weddwitt.html#E30:52"><i><font color="Green" title="E61">A32</font></i></a>, <a class="txt" href="weddwitt.html#E31:52"><i><font color="Green" title="E62">A33</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c18">p1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E37:52"/><i><font color="Green" title="E64">A36</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">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="c15">f1</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="c15">f1</font></span>)</span> holds <br/><font color="Maroon" title="c18">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="E38:52"/><i><font color="Green" title="E65">A37</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c15">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E39:52"/>
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="c18">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#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_6"/><i><font color="Green" title="E66">A38</font></i>: 
<font color="Maroon" title="c19">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="c18">p1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c20">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="E3:52_6"/><i><font color="Green" title="E67">A39</font></i>: 
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">p1</font> &amp; <font color="Maroon" title="c18">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:52_6"><i><font color="Green" title="E66">A38</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:52_6"/><i><font color="Green" title="E68">A40</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E38:52"><i><font color="Green" title="E65">A37</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_6"><i><font color="Green" title="E67">A39</font></i></a></i>;<br/>
<a NAME="E5:52_6"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">f1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_6"><i><font color="Green" title="E67">A39</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:52_6"/><b>then </b>
<font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E26:52"><i><font color="Green" title="E58">A29</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E8:52_6"/><i><font color="Green" title="E69">A41</font></i>: 
( <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">X</font> &amp; <font color="Maroon" title="c21">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c21">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E9:52_6"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:52_6"><i><font color="Green" title="E68">A40</font></i></a>, <a class="txt" href="weddwitt.html#E8:52_6"><i><font color="Green" title="E69">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:52"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c19">c1</font> = <font color="Maroon" title="c18">p1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="E42:52"/><i><font color="Green" title="E66">A42</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">natq1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E23:52"><i><font color="Green" title="E57">A28</font></i></a>, <a class="txt" href="weddwitt.html#E26:52"><i><font color="Green" title="E58">A29</font></i></a>, <a class="txt" href="weddwitt.html#E27:52"><i><font color="Green" title="E59">A30</font></i></a>, <a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E43:52"/><i><font color="Green" title="E67">A43</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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font> holds <br/><font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">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="E1:52_7"/><i><font color="Green" title="E68">A44</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font>
 ;<br/>

<a NAME="E2:52_7"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">f1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E1:52_7"><i><font color="Green" title="E68">A44</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:52_7"/><b>then </b>
<font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E26:52"><i><font color="Green" title="E58">A29</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:52_7"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> st <br/>( <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
;<br/>
<b>hence </b><a NAME="E5:52_7"/>
<font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E38:52"><i><font color="Green" title="E65">A37</font></i></a>, <a class="txt" href="weddwitt.html#E1:52_7"><i><font color="Green" title="E68">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:52"/>
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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_8"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">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="E3:52_8"/><i><font color="Green" title="E68">A45</font></i>: 
( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font> &amp; <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c21">i</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:52_8"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E43:52"><i><font color="Green" title="E67">A43</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_8"><i><font color="Green" title="E68">A45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:52_8"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:52"/><b>then </b><i><font color="Green" title="E68">A46</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E46:52"/>
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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_9"/><i><font color="Green" title="E69">A47</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:52_9"/>
 <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="c19">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:52_9"/><b>then </b><i><font color="Green" title="E70">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:52"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="weddwitt.html#E42:52"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:52_9"/><b>then </b>
<font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E43:52"><i><font color="Green" title="E67">A43</font></i></a></i>;<br/>
<a NAME="E5:52_9"/><b>then </b>
<font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:52_9"><i><font color="Green" title="E69">A47</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:52_9"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:52_9"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:52"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E48:52"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E45:52"><i><font color="Green" title="E68">A46</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E49:52"/><b>then </b>
<font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a></i>;<br/>
<a NAME="E50:52"/><b>then </b>
<font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E51:52"/><b>then </b>
<font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#D2" title="FINSEQ_2:def.2">FINSEQ_2:def 2</a></i>;<br/>
<a NAME="E52:52"/><b>then </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>;<br/>
<a NAME="E53:52"/><b>then </b><i><font color="Green" title="E69">A49</font></i>: 
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c19">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">natq1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E23:52"><i><font color="Green" title="E57">A28</font></i></a>, <a class="txt" href="weddwitt.html#E26:52"><i><font color="Green" title="E58">A29</font></i></a>, <a class="txt" href="weddwitt.html#E27:52"><i><font color="Green" title="E59">A30</font></i></a>, <a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c20">p2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E55:52"/><i><font color="Green" title="E70">A50</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">p2</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="c16">f2</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="c16">f2</font></span>)</span> holds <br/><font color="Maroon" title="c20">p2</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>2</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="E56:52"/><i><font color="Green" title="E71">A51</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c16">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E57:52"/>
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="c20">p2</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#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_10"/>
<font color="Maroon" title="c21">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="c20">p2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">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="E3:52_10"/><i><font color="Green" title="E72">A52</font></i>: 
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">p2</font> &amp; <font color="Maroon" title="c20">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:52_10"/><i><font color="Green" title="E73">A53</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="weddwitt.html#E56:52"><i><font color="Green" title="E71">A51</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_10"><i><font color="Green" title="E72">A52</font></i></a></i>;<br/>
<a NAME="E5:52_10"/>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">f2</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_10"><i><font color="Green" title="E72">A52</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:52_10"/><b>then </b>
<font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E29:52"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:52_10"/><b>then </b>
<font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">a</font> being    <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> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E9:52_10"/><i><font color="Green" title="E74">A54</font></i>: 
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T14" target="_self" title="WEDDWITT:th.14">Th14</a></i>;<br/>
<a NAME="E10:52_10"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c23">a</font></span>)</span> is    <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>hence </b><a NAME="E11:52_10"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:52_10"><i><font color="Green" title="E73">A53</font></i></a>, <a class="txt" href="weddwitt.html#E9:52_10"><i><font color="Green" title="E74">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E58:52"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c21">c2</font> = <font color="Maroon" title="c20">p2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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/>
<b>set </b><font color="Maroon" title="c22">c</font> = <font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <font color="Maroon" title="c21">c2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">c</font> = <font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <font color="Maroon" title="c21">c2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E61:52"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E62:52"/><b>then </b><i><font color="Green" title="E72">A55</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E63:52"/>
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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">c</font> holds <br/><font color="Maroon" title="c23">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">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="E1:52_11"/><i><font color="Green" title="E73">A56</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">c</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_11_1_1"/>
( <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font> or  ex <font color="Olive" title="b1">j</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 <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c2</font> &amp; <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><b>by </b><i><a class="txt" href="weddwitt.html#E1:52_11"><i><font color="Green" title="E73">A56</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_11_1_1_1"><b>suppose </b></a><a NAME="E1:52_11_1_1_1"/><i><font color="Green" title="E74">A57</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c1</font>
 ;<br/><div class="add"><a NAME="E2:52_11_1_1_1"/><b>then </b><i><font color="Green" title="E75">A58</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">f1</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><font color="Maroon" title="c23">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c24">i</font> = 
<font color="Maroon" title="c19">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c24">i</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E1:52_11_1_1_1"><i><font color="Green" title="E74">A57</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E38:52"><i><font color="Green" title="E65">A37</font></i></a>, <a class="txt" href="weddwitt.html#E1:52_11_1_1_1"><i><font color="Green" title="E74">A57</font></i></a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:52_11_1_1_1"><i><font color="Green" title="E75">A58</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E4:52_11_1_1_1"/>
<font color="Maroon" title="c23">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_11_1_1_2"><b>suppose </b></a><a NAME="E1:52_11_1_1_2"/>
 ex <font color="Olive" title="b1">j</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 <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c2</font> &amp; <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">j</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c25">j</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="E3:52_11_1_1_2"/><i><font color="Green" title="E74">A59</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c2</font>
 <b>and </b><br/><a NAME="E4:52_11_1_1_2"/><i><font color="Green" title="E75">A60</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">j</font>
 ;<br/><a NAME="E5:52_11_1_1_2"/><i><font color="Green" title="E76">A61</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">f2</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_11_1_1_2"><i><font color="Green" title="E74">A59</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><font color="Maroon" title="c23">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c24">i</font> = 
<font color="Maroon" title="c21">c2</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c25">j</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E3:52_11_1_1_2"><i><font color="Green" title="E74">A59</font></i></a>, <a class="txt" href="weddwitt.html#E4:52_11_1_1_2"><i><font color="Green" title="E75">A60</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="weddwitt.html#E56:52"><i><font color="Green" title="E71">A51</font></i></a>, <a class="txt" href="weddwitt.html#E3:52_11_1_1_2"><i><font color="Green" title="E74">A59</font></i></a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E37:52"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="weddwitt.html#E4:52_11_1_1_2"><i><font color="Green" title="E75">A60</font></i></a>, <a class="txt" href="weddwitt.html#E5:52_11_1_1_2"><i><font color="Green" title="E76">A61</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E7:52_11_1_1_2"/>
<font color="Maroon" title="c23">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_11"/>
<font color="Maroon" title="c23">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E64:52"/><b>then </b>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c23">c</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E31:52"><i><font color="Green" title="E62">A33</font></i></a>, <a class="txt" href="weddwitt.html#E35:52"><i><font color="Green" title="E63">A35</font></i></a>, <a class="txt" href="weddwitt.html#E62:52"><i><font color="Green" title="E72">A55</font></i></a>, <a class="ref" href="weddwitt.html#T15" target="_self" title="WEDDWITT:th.15">Th15</a></i>;<br/>
<a NAME="E65:52"/><b>then </b><i><font color="Green" title="E73">A62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c21">c2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="weddwitt.html#E53:52"><i><font color="Green" title="E69">A49</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">q</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" title="WEDDWITT:func.4">center</a> <font color="Maroon" title="c1">R</font></span>)</span> as  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>  <b>by </b><i><a class="ref" href="weddwitt.html#T21" target="_self" title="WEDDWITT:th.21">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">n</font> =  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" title="WEDDWITT:func.6">VectSp_over_center</a> <font color="Maroon" title="c1">R</font></span>)</span> as  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>  <b>by </b><i><a class="ref" href="weddwitt.html#T33" target="_self" title="WEDDWITT:th.33">Th33</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E68:52"/>
<font color="Maroon" title="c24">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="c26">qc</font> = <font color="Maroon" title="c24">q</font> 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>set </b><font color="Maroon" title="c27">pnq</font> =  <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="c25">n</font></span>)</span>,<font color="Maroon" title="c26">qc</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">pnq</font> =  <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="c25">n</font></span>)</span>,<font color="Maroon" title="c26">qc</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="uniroots.html#T56" title="UNIROOTS:th.56">UNIROOTS:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">abspnq</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c28">pnq</font> as    <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>reconsider </b><font color="Maroon" title="c30">nataq1</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> as    <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/>
<a NAME="E73:52"/>
<font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<a NAME="E74:52"/><b>then </b>
<font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E75:52"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E76:52"/><b>then </b>
<font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E77:52"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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/>
<b>then reconsider </b><font color="Maroon" title="c31">qn1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E79:52"/>
<font color="Maroon" title="c28">pnq</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="uniroots.html#T62" title="UNIROOTS:th.62">UNIROOTS:62</a></i>;<br/>
<a NAME="E80:52"/><b>then </b>
<font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></i>;<br/>
<a NAME="E81:52"/><b>then </b><i><font color="Green" title="E74">A63</font></i>: 
<font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c31">qn1</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E82:52"/>
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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c2</font> holds <br/><font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c21">c2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">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="E1:52_12"/><i><font color="Green" title="E75">A64</font></i>: 
<font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c2</font>
 ;<br/>

<a NAME="E2:52_12"/>
<font color="Maroon" title="c21">c2</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="weddwitt.html#E56:52"><i><font color="Green" title="E71">A51</font></i></a>, <a class="txt" href="weddwitt.html#E1:52_12"><i><font color="Green" title="E75">A64</font></i></a></i>;<br/>
<a NAME="E3:52_12"/><b>then </b><i><font color="Green" title="E76">A65</font></i>: 
<font color="Maroon" title="c21">c2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:52_12"><i><font color="Green" title="E75">A64</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E4:52_12"/><i><font color="Green" title="E77">A66</font></i>: 
<font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">f2</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E55:52"><i><font color="Green" title="E70">A50</font></i></a>, <a class="txt" href="weddwitt.html#E1:52_12"><i><font color="Green" title="E75">A64</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:52_12"/><b>then </b>
<font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E29:52"><i><font color="Green" title="E60">A31</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:52_12"/><b>then </b>
( <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp; not <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c33">a</font> being    <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> <font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E8:52_12"/><i><font color="Green" title="E78">A67</font></i>: 
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c33">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T14" target="_self" title="WEDDWITT:th.14">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">a</font> = <font color="Maroon" title="c33">a</font> as   <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> <font color="Maroon" title="c1">R</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c35">s</font> = <font color="Maroon" title="c34">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="uniroots.html#T22" title="UNIROOTS:th.22">UNIROOTS:22</a></i>;<br/>
<b>set </b><font color="Maroon" title="c36">ns</font> =  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c37">ca</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c34">a</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c38">oa</font> =  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c34">a</font></span>)</span>;<br/>
<a NAME="E11:52_12"/><i><font color="Green" title="E79">A68</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c34">a</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c34">a</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 
<b>by </b><i><a class="ref" href="weddwitt.html#T13" target="_self" title="WEDDWITT:th.13">Th13</a></i>;<br/>
<a NAME="E12:52_12"/><i><font color="Green" title="E80">A69</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c34">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E13:52_12"/><b>then </b><i><font color="Green" title="E81">A70</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c34">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c34">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E11:52_12"><i><font color="Green" title="E79">A68</font></i></a>, <a class="ref" href="nat_d.html#D1" title="NAT_D:def.1">NAT_D:def 1</a></i>;<br/>
<a NAME="E14:52_12"/><i><font color="Green" title="E82">A71</font></i>: 
<font color="Maroon" title="c31">qn1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c34">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c34">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="weddwitt.html#E11:52_12"><i><font color="Green" title="E79">A68</font></i></a>, <a class="txt" href="weddwitt.html#E12:52_12"><i><font color="Green" title="E80">A69</font></i></a>, <a class="ref" href="nat_d.html#D1" title="NAT_D:def.1">NAT_D:def 1</a></i>;<br/>
<a NAME="E15:52_12"/>
<font color="Maroon" title="c24">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<a NAME="E16:52_12"/><b>then </b>
<font color="Maroon" title="c24">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E17:52_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E18:52_12"/><b>then </b>
<font color="Maroon" title="c24">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E19:52_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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/>
<b>then reconsider </b><font color="Maroon" title="c39">qns1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<i><font color="Green" title="E83">A72</font></i>:  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c34">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K5" title="WEDDWITT:func.5">centralizer</a> <font color="Maroon" title="c35">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="weddwitt.html#T31" target="_self" title="WEDDWITT:th.31">Th31</a></i>
<br/>.= 
<font color="Maroon" title="c39">qns1</font>
<b>by </b><i><a class="ref" href="weddwitt.html#T34" target="_self" title="WEDDWITT:th.34">Th34</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c40">ns</font> =  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K7" title="WEDDWITT:func.7">VectSp_over_center</a> <font color="Maroon" title="c35">s</font></span>)</span> as  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>  <b>by </b><i><a class="ref" href="weddwitt.html#T35" target="_self" title="WEDDWITT:th.35">Th35</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E23:52_12"/><i><font color="Green" title="E84">A73</font></i>: 
<font color="Maroon" title="c40">ns</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c25">n</font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T37" target="_self" title="WEDDWITT:th.37">Th37</a></i>;<br/>
<a NAME="E24:52_12"/><b>then </b><i><font color="Green" title="E85">A74</font></i>: 
<font color="Maroon" title="c40">ns</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_12_2"/><i><font color="Green" title="E86">A75</font></i>: 
<font color="Maroon" title="c40">ns</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">n</font>
 ;<br/><a NAME="E2:52_12_2"/>
<font color="Maroon" title="c31">qn1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:52"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="pepin.html#T26" title="PEPIN:th.26">PEPIN:26</a></i>;<br/><a NAME="E3:52_12_2"/><b>then </b><i><font color="Green" title="E87">A76</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="weddwitt.html#E8:52_12"><i><font color="Green" title="E78">A67</font></i></a>, <a class="txt" href="weddwitt.html#E13:52_12"><i><font color="Green" title="E81">A70</font></i></a>, <a class="txt" href="weddwitt.html#E21:52_12"><i><font color="Green" title="E83">A72</font></i></a>, <a class="txt" href="weddwitt.html#E1:52_12_2"><i><font color="Green" title="E86">A75</font></i></a>, <a class="ref" href="nat_2.html#T5" title="NAT_2:th.5">NAT_2:5</a></i>;<br/><a NAME="E4:52_12_2"/>
<font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E29:52"><i><font color="Green" title="E60">A31</font></i></a>, <a class="txt" href="weddwitt.html#E4:52_12"><i><font color="Green" title="E77">A66</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E5:52_12_2"/><b>then </b>
( <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp; not <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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> <font color="Maroon" title="c1">R</font></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="weddwitt.html#K3" title="WEDDWITT:func.3">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:52_12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:52_12_2"><i><font color="Green" title="E87">A76</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:52_12"/><b>then </b>
<font color="Maroon" title="c40">ns</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c25">n</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E24:52_12"><i><font color="Green" title="E85">A74</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E27:52_12"/><b>then </b>
<font color="Maroon" title="c28">pnq</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c31">qn1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c39">qns1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E23:52_12"><i><font color="Green" title="E84">A73</font></i></a>, <a class="ref" href="uniroots.html#T63" title="UNIROOTS:th.63">UNIROOTS:63</a></i>;<br/>
<a NAME="E28:52_12"/><b>then </b>
<font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">qn1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c39">qns1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E29:52_12"/>
<font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c21">c2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c32">i</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:52_12"><i><font color="Green" title="E76">A65</font></i></a>, <a class="txt" href="weddwitt.html#E8:52_12"><i><font color="Green" title="E78">A67</font></i></a>, <a class="txt" href="weddwitt.html#E14:52_12"><i><font color="Green" title="E82">A71</font></i></a>, <a class="txt" href="weddwitt.html#E21:52_12"><i><font color="Green" title="E83">A72</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E83:52"/><b>then </b>
<font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c21">c2</font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T5" target="_self" title="WEDDWITT:th.5">Th5</a></i>;<br/>
<a NAME="E84:52"/><b>then </b><i><font color="Green" title="E75">A77</font></i>: 
<font color="Maroon" title="c29">abspnq</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">natq1</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E65:52"><i><font color="Green" title="E73">A62</font></i></a>, <a class="txt" href="weddwitt.html#E81:52"><i><font color="Green" title="E74">A63</font></i></a>, <a class="ref" href="nat_d.html#T10" title="NAT_D:th.10">NAT_D:10</a></i>;<br/>
<a NAME="E85:52"/>
<font color="Maroon" title="c29">abspnq</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:52"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="weddwitt.html#E11:52"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="uniroots.html#T64" title="UNIROOTS:th.64">UNIROOTS:64</a></i>;<br/>
<b>hence </b><a NAME="E86:52"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:52"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="weddwitt.html#E84:52"><i><font color="Green" title="E75">A77</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div>
