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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> : verum </span>}</span>  as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="weddwitt.html#T11" target="_self" title="WEDDWITT:th.11">Th11</a></i>;<br/>
<a NAME="E2:19"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; ( for <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="weddwitt.html#E1:19_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:19_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:19_1"/>
( <font color="Maroon" title="c4">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; ( for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c4">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">B</font> ) )
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:19_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="weddwitt.html#E4:19_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="weddwitt.html#T9" target="_self" title="WEDDWITT:th.9">Th9</a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c4">k</font> =  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:19"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; ( for <font color="Olive" title="b3">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">Z</font> holds <br/>( <font color="Olive" title="b1">Y</font>,<font color="Olive" title="b3">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Olive" title="b1">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Z</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c5">Y</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E2:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="weddwitt.html#E1:19_2"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E3:19_2"/><i><font color="Green" title="E18">A5</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="weddwitt.html#E1:19_2"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E4:19_2"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">Z</font> holds <br/>( <font color="Maroon" title="c6">Y</font>,<font color="Olive" title="b1">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c6">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19_2_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c7">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E2:19_2_1"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">Z</font>
 ;<br/>

<a NAME="E3:19_2_1"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="weddwitt.html#E1:19_2"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E4:19_2_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="weddwitt.html#E1:19_2_1"><i><font color="Green" title="E19">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:19_2_1"/>
( <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c7">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c6">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Z</font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="weddwitt.html#E1:19_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="weddwitt.html#E1:19_2_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="weddwitt.html#E2:19_2_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="weddwitt.html#E3:19_2_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_2"/>
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; ( for <font color="Olive" title="b2">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">Z</font> holds <br/>( <font color="Maroon" title="c5">Y</font>,<font color="Olive" title="b2">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c5">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">Z</font> ) ) )
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:19_2"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:19"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E7:19"/><i><font color="Green" title="E18">A10</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="group_2.html#T186" title="GROUP_2:th.186">GROUP_2:186</a></i>;<br/>
 <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">C</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E6:19"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E7:19"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="weddwitt.html#T12" target="_self" title="WEDDWITT:th.12">Th12</a></i>
;<br/>
<b>hence </b><a NAME="E9:19"/>
 <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p2">(<span class="default"><a href="weddwitt.html#K1" title="WEDDWITT:func.1">Centralizer</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 ;<br/>


</div>
