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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:142"/><i><font color="Green">E38</font></i>: 
(  <a href="group_3.html#K8" target="_self">con_class</a> <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> or  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<i><font color="Green">E39</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_3.html#K8" target="_self">con_class</a> <font color="Maroon">A</font></span>)</span> = 
 <a href="group_2.html#K16">Index</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T65" target="_self">Th65</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_2.html#K14">Left_Cosets</a> <span class="p2">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span></span>)</span>

;<br/>
<a NAME="E3:142"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="group_3.html#K8" target="_self">con_class</a> <font color="Maroon">A</font>, <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E4:142"/><b>then </b>
(  <a href="group_3.html#K8" target="_self">con_class</a> <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="group_3.html#D6" target="_self">Def6</a>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:142"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="group_2.html#K17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="group_2.html#D18">GROUP_2:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="group_3.html#K8" target="_self">con_class</a> <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_3.html#D6" target="_self">Def6</a>, <a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E8:142"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K8" target="_self">con_class</a> <font color="Maroon">A</font>
 ;<br/>

<b>thus </b><a NAME="E9:142"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K17">index</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" target="_self">Normalizator</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#T70" target="_self">Th70</a>, <a class="ref" href="group_3.html#T72" target="_self">Th72</a></i>;<br/>


</div>
