<?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">H</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:139"/><i><font color="Green">E35</font></i>: 
 <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> =  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><b>hereby </b>
<div class="add"><b>take </b><font color="Maroon">B</font> = <font color="Maroon">B</font>;<br/><b>thus </b><a NAME="E1:139_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<a NAME="E4:139"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:139"/><i><font color="Green">E47</font></i>: 
 <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font>, <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</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="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E7:139"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K17" target="_self">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="group_2.html#T53" target="_self">Th53</a>, <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/>


</div>
