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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="vectsp_2.html#NM3">Skew-Field</a>;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V7">commutative</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E2:53"/><i><font color="Green">E47</font></i>: 
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E3:53"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="uniroots.html#T21">UNIROOTS:21</a></i>;<br/>
<a NAME="E4:53"/><i><font color="Green">E49</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E50</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E51</font></i>: 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:53"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E8:53"/><i><font color="Green">E52</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E9:53"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:53"/><b>then </b><i><font color="Green">E53</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E54</font></i>: 
 <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:53_1"/>
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="prepower.html#K7">#Z</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>;<br/><a NAME="E3:53_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:53_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>;<br/><b>hence </b><a NAME="E4:53_1"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:53"><i><font color="Green">E46</font></i></a>, <a class="ref" href="weddwitt.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:53"/><b>then </b><i><font color="Green">E54</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E10:53"><i><font color="Green">E53</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<a NAME="E13:53"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 ;<br/>

<a NAME="E2:53_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
;<br/>
<b>hence </b><a NAME="E3:53_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:53"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:53"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p3">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:53"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E18:53"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E19:53"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:53_3"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/>

<a NAME="E2:53_3"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E18:53"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_4_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_4_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:53_4_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:53_4_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E18:53"><i><font color="Green">E58</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_4_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="weddwitt.html#E4:53_4_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E6:53_4_1"/><i><font color="Green">E63</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_4_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="weddwitt.html#E5:53_4_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_4_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="weddwitt.html#E6:53_4_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E9:53_4_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_5.html#K10">center</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_4_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E10:53_4_1"/><b>then </b>
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="group_5.html#T93">GROUP_5:93</a></i>;<br/><a NAME="E11:53_4_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:53_4_1"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E12:53_4_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:53_4_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><b>hence </b><a NAME="E13:53_4_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E11:53_4_1"><i><font color="Green">E64</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:53_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:53_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:53_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:53_4"/><i><font color="Green">E62</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E8:53_4"/><i><font color="Green">E63</font></i>: 
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:53_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="weddwitt.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E9:53_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#T98">GROUP_3:98</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:53_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:53_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E12:53_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:53_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="weddwitt.html#E9:53_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="weddwitt.html#E11:53_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:53_4"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_5.html#K10">center</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:53_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="weddwitt.html#E11:53_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="group_5.html#T93">GROUP_5:93</a></i>;<br/><a NAME="E14:53_4"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E15:53_4"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E18:53"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E16:53_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E4:53_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="weddwitt.html#E11:53_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="weddwitt.html#E12:53_4"><i><font color="Green">E66</font></i></a>, <a class="txt" href="weddwitt.html#E14:53_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:53"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E22:53"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E17:53"><i><font color="Green">E57</font></i></a>, <a class="txt" href="weddwitt.html#E19:53"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E23:53"/>
 <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_5.html#K10">center</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E24:53"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E22:53"><i><font color="Green">E60</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E25:53"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E14:53"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E27:53"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 <b>and </b><br/><a NAME="E28:53"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E30:53"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 <b>and </b><br/><a NAME="E31:53"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font>;<br/>
<a NAME="E32:53"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E15:53"><i><font color="Green">E56</font></i></a>, <a class="txt" href="weddwitt.html#E27:53"><i><font color="Green">E62</font></i></a>, <a class="txt" href="weddwitt.html#E30:53"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:53_5"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p3">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:53_5"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:53_5"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:53_5"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E34:53"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p3">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E35:53"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:53"><i><font color="Green">E62</font></i></a>, <a class="txt" href="weddwitt.html#E30:53"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E36:53"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font> is  <a href="funct_1.html#V2">one-to-one</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E28:53"><i><font color="Green">E63</font></i></a>, <a class="txt" href="weddwitt.html#E31:53"><i><font color="Green">E65</font></i></a>, <a class="txt" href="weddwitt.html#E32:53"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E38:53"/><i><font color="Green">E68</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E39:53"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E40:53"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_6"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53_6"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:53_6"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:53_6"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E39:53"><i><font color="Green">E69</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_6"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E5:53_6"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_6"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:53_6"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:53"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E8:53_6"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E9:53_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:53_6"><i><font color="Green">E72</font></i></a>, <a class="txt" href="weddwitt.html#E8:53_6"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E41:53"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E43:53"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E24:53"><i><font color="Green">E61</font></i></a>, <a class="txt" href="weddwitt.html#E27:53"><i><font color="Green">E62</font></i></a>, <a class="txt" href="weddwitt.html#E28:53"><i><font color="Green">E63</font></i></a>, <a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E44:53"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:53_7"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E2:53_7"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E1:53_7"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:53_7"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:53"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:53_7"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
;<br/>
<b>hence </b><a NAME="E5:53_7"/>
<font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E39:53"><i><font color="Green">E69</font></i></a>, <a class="txt" href="weddwitt.html#E1:53_7"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:53"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_8"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53_8"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:53_8"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E44:53"><i><font color="Green">E71</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_8"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:53_8"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:53"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E47:53"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_9"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:53_9"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:53_9"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:53"><i><font color="Green">E51</font></i></a>, <a class="txt" href="weddwitt.html#E43:53"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:53_9"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E44:53"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E5:53_9"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:53_9"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:53_9"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:53_9"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:53"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E49:53"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E46:53"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E50:53"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a></i>;<br/>
<a NAME="E51:53"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E52:53"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#D2">FINSEQ_2:def 2</a></i>;<br/>
<a NAME="E53:53"/><b>then </b>
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>;<br/>
<a NAME="E54:53"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E24:53"><i><font color="Green">E61</font></i></a>, <a class="txt" href="weddwitt.html#E27:53"><i><font color="Green">E62</font></i></a>, <a class="txt" href="weddwitt.html#E28:53"><i><font color="Green">E63</font></i></a>, <a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E56:53"/><i><font color="Green">E74</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E57:53"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E58:53"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_10"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53_10"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:53_10"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="txt" href="weddwitt.html#E57:53"><i><font color="Green">E75</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_10"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E5:53_10"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_10"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:53_10"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E30:53"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:53_10"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:53_10"/><i><font color="Green">E78</font></i>: 
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E10:53_10"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> is   <a href="nat_1.html#NM1">Nat</a>
 
;<br/>
<b>hence </b><a NAME="E11:53_10"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:53_10"><i><font color="Green">E77</font></i></a>, <a class="txt" href="weddwitt.html#E9:53_10"><i><font color="Green">E78</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:53"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">c<sub>21</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">c<sub>21</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E62:53"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E63:53"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E64:53"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:53_11"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_11_1_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="weddwitt.html#E1:53_11"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_11_1_1_1"><b>suppose </b></a><a NAME="E1:53_11_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:53_11_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><font color="Maroon">c<sub>23</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="weddwitt.html#E1:53_11_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E39:53"><i><font color="Green">E69</font></i></a>, <a class="txt" href="weddwitt.html#E1:53_11_1_1_1"><i><font color="Green">E78</font></i></a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E2:53_11_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E4:53_11_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_11_1_1_2"><b>suppose </b></a><a NAME="E1:53_11_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53_11_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:53_11_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><a NAME="E5:53_11_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_11_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><font color="Maroon">c<sub>23</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="weddwitt.html#E3:53_11_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="weddwitt.html#E4:53_11_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="txt" href="weddwitt.html#E57:53"><i><font color="Green">E75</font></i></a>, <a class="txt" href="weddwitt.html#E3:53_11_1_1_2"><i><font color="Green">E78</font></i></a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E38:53"><i><font color="Green">E68</font></i></a>, <a class="txt" href="weddwitt.html#E4:53_11_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="weddwitt.html#E5:53_11_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E7:53_11_1_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:53_11"/>
<font color="Maroon">c<sub>23</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E65:53"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E32:53"><i><font color="Green">E66</font></i></a>, <a class="txt" href="weddwitt.html#E36:53"><i><font color="Green">E67</font></i></a>, <a class="txt" href="weddwitt.html#E63:53"><i><font color="Green">E76</font></i></a>, <a class="ref" href="weddwitt.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E66:53"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E67:53"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>21</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E68:53"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:53"><i><font color="Green">E48</font></i></a>, <a class="txt" href="weddwitt.html#E54:53"><i><font color="Green">E73</font></i></a>, <a class="txt" href="weddwitt.html#E65:53"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> =  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K4" target="_self">center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="weddwitt.html#E5:53"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K6" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="weddwitt.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E71:53"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>27</sub></font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6">cyclotomic_poly</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> =  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K6">cyclotomic_poly</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="uniroots.html#T56">UNIROOTS:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> =  <a href="prepower.html#K1">abs</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> =  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E76:53"/>
<font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E77:53"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E78:53"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E79:53"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E80:53"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E82:53"/>
<font color="Maroon">c<sub>28</sub></font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="uniroots.html#T62">UNIROOTS:62</a></i>;<br/>
<a NAME="E83:53"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T21">INT_2:21</a></i>;<br/>
<a NAME="E84:53"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E80:53"><i><font color="Green">E79</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E85:53"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> holds <br/><font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:53_12"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<a NAME="E2:53_12"/>
<font color="Maroon">c<sub>21</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="txt" href="weddwitt.html#E57:53"><i><font color="Green">E75</font></i></a>, <a class="txt" href="weddwitt.html#E1:53_12"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E3:53_12"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:53_12"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E4:53_12"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E56:53"><i><font color="Green">E74</font></i></a>, <a class="txt" href="weddwitt.html#E1:53_12"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:53_12"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E30:53"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:53_12"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E8:53_12"/><i><font color="Green">E84</font></i>: 
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="ref" href="weddwitt.html#T14" target="_self">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>34</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="uniroots.html#T22">UNIROOTS:22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>36</sub></font> =  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>37</sub></font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>38</sub></font> =  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>;<br/>
<a NAME="E11:53_12"/><i><font color="Green">E85</font></i>: 
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K9">ord</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="weddwitt.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E12:53_12"/>
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E13:53_12"/><b>then </b><i><font color="Green">E86</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E14:53_12"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E11:53_12"><i><font color="Green">E85</font></i></a>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>
<a NAME="E15:53_12"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:53"><i><font color="Green">E48</font></i></a>, <a class="txt" href="weddwitt.html#E11:53_12"><i><font color="Green">E85</font></i></a>, <a class="txt" href="weddwitt.html#E13:53_12"><i><font color="Green">E86</font></i></a>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>
<a NAME="E16:53_12"/>
<font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E17:53_12"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E18:53_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E19:53_12"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E20:53_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>39</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <span class="p3">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<i><font color="Green">E89</font></i>:  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="weddwitt.html#K5" target="_self">centralizer</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="weddwitt.html#T31" target="_self">Th31</a></i>
<br/>.= 
<font color="Maroon">c<sub>39</sub></font>
<b>by </b><i><a class="ref" href="weddwitt.html#T34" target="_self">Th34</a></i>
;<br/>
<a NAME="E23:53_12"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E24:53_12"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>39</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>40</sub></font> =  <a href="vectsp_9.html#K1">dim</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K7" target="_self">VectSp_over_center</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="weddwitt.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E26:53_12"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E27:53_12"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E8:53"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_12_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><a NAME="E2:53_12_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:53"><i><font color="Green">E49</font></i></a>, <a class="txt" href="weddwitt.html#E8:53"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pepin.html#T26">PEPIN:26</a></i>;<br/><a NAME="E3:53_12_2"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:53"><i><font color="Green">E48</font></i></a>, <a class="txt" href="weddwitt.html#E8:53_12"><i><font color="Green">E84</font></i></a>, <a class="txt" href="weddwitt.html#E14:53_12"><i><font color="Green">E87</font></i></a>, <a class="txt" href="weddwitt.html#E22:53_12"><i><font color="Green">E89</font></i></a>, <a class="txt" href="weddwitt.html#E1:53_12_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="nat_2.html#T5">NAT_2:5</a></i>;<br/><a NAME="E4:53_12_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p2">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="weddwitt.html#E30:53"><i><font color="Green">E64</font></i></a>, <a class="txt" href="weddwitt.html#E4:53_12"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:53_12_2"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="weddwitt.html#K3" target="_self">conjugate_Classes</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1">MultGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:53_12_2"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:53_12_2"><i><font color="Green">E95</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E29:53_12"/><b>then </b>
<font color="Maroon">c<sub>40</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E27:53_12"><i><font color="Green">E93</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E30:53_12"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>31</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E26:53_12"><i><font color="Green">E92</font></i></a>, <a class="ref" href="uniroots.html#T63">UNIROOTS:63</a></i>;<br/>
<a NAME="E31:53_12"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E23:53_12"><i><font color="Green">E90</font></i></a>, <a class="ref" href="int_2.html#T21">INT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E32:53_12"/>
<font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:53_12"><i><font color="Green">E82</font></i></a>, <a class="txt" href="weddwitt.html#E8:53_12"><i><font color="Green">E84</font></i></a>, <a class="txt" href="weddwitt.html#E15:53_12"><i><font color="Green">E88</font></i></a>, <a class="txt" href="weddwitt.html#E22:53_12"><i><font color="Green">E89</font></i></a>, <a class="txt" href="weddwitt.html#E24:53_12"><i><font color="Green">E91</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E86:53"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E87:53"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E68:53"><i><font color="Green">E78</font></i></a>, <a class="txt" href="weddwitt.html#E84:53"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#T57">NAT_1:57</a></i>;<br/>
<a NAME="E88:53"/>
<font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>24</sub></font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:53"><i><font color="Green">E49</font></i></a>, <a class="txt" href="weddwitt.html#E12:53"><i><font color="Green">E54</font></i></a>, <a class="ref" href="uniroots.html#T64">UNIROOTS:64</a></i>;<br/>
<b>hence </b><a NAME="E89:53"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:53"><i><font color="Green">E51</font></i></a>, <a class="txt" href="weddwitt.html#E87:53"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div>
