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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">D</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:22"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B</font> &amp; <font color="Maroon">D</font> <a href="group_2.html#R1">=</a> <font color="Maroon">A</font> <a href="group_2.html#K9">/\</a> <font color="Maroon">B</font> &amp; <font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">LCB</font> =  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_2.html#T164">GROUP_2:164</a></i>;<br/>
<a NAME="E3:22"/>
<font color="Maroon">A</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LCD</font> =  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_2.html#T164">GROUP_2:164</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD</font> = <font color="Maroon">D</font> as    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> ;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">x'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/><a NAME="E2:22_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y'</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K3">"</a> 
 <b>by </b><i>, <a class="ref" href="group_2.html#T57">GROUP_2:57</a></i>;<br/><a NAME="E3:22_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:22_1"/>
( <font color="Maroon">x</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> iff <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T137">GROUP_2:137</a></i>;<br/><a NAME="E5:22_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> iff <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/><a NAME="E6:22_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> iff <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/><b>hence </b><a NAME="E7:22_1"/>
( <font color="Maroon">x</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> iff <font color="Maroon">x'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T137">GROUP_2:137</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ex <font color="Olive">a'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font> );<br/>
<a NAME="E7:22"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">LCD</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">LCB</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">LCD</font>;<br/>

<a NAME="E1:22_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LCD</font>
 
;<br/>
<b>then consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">LCB</font> <b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">a'</font>
;<br/>

<b>thus </b><a NAME="E6:22_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">LCD</font>,<font color="Maroon">LCB</font><b> such that </b><br/><a NAME="E9:22"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">LCD</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E10:22"/><i><font color="Green">E64</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCD</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">LCB</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E65</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">LCD</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D18">GROUP_2:def 18</a></i>;<br/>
<a NAME="E12:22"/><i><font color="Green">E68</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">LCB</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D18">GROUP_2:def 18</a></i>;<br/>
<a NAME="E13:22"/>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</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="E1:22_3"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">x1</font>, <font color="Maroon">z2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">LCD</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:22_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:22_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="group_2.html#K12">*</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:22_3"/>
(  not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> or <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:22"/><b>then </b>
 <a href="group_2.html#K17">index</a> <font color="Maroon">D</font> <a href="ordinal1.html#R1">c=</a>  <a href="group_2.html#K17">index</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , , <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<b>hence </b><a NAME="E15:22"/>
 <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">G</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="group_8.html#NK2" target="_self">index</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>


</div>
