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

<b>let </b><font color="Maroon" title="c1">G</font> be   <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>let </b><font color="Maroon" title="c3">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> );<br/>
<a NAME="E1:69"/><i><font color="Green" title="E50">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>set </b><font color="Maroon" title="c7">A</font> =  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font>;<br/>
<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E3:69_1"/><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 ;<br/>

<b>given </b><font color="Maroon" title="c9">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E5:69_1"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 ;<br/>

<b>thus </b><font color="Maroon" title="c5">y1</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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="txt" href="group_3.html#E3:69_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="group_3.html#T71" target="_self" title="GROUP_3:th.71">Th71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T39" title="GROUP_2:th.39">GROUP_2:39</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p4">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="group_3.html#E3:69_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="group_3.html#E5:69_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p4">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>
<br/>.= 
<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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <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="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T39" title="GROUP_2:th.39">GROUP_2:39</a></i>
<br/>.= 
<font color="Maroon" title="c6">y2</font>
<b>by </b><i><a class="txt" href="group_3.html#E5:69_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="group_3.html#T71" target="_self" title="GROUP_3:th.71">Th71</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:69"/><i><font color="Green" title="E51">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:69_2"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c6">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">b</font>
;<br/>

<b>thus </b><a NAME="E5:69_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_3.html#E3:69_2"><i><font color="Green" title="E52">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:69"/><i><font color="Green" title="E52">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font>
 <b>and </b><br/><a NAME="E5:69"/><i><font color="Green" title="E53">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="group_3.html#E1:69"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="group_3.html#E2:69"><i><font color="Green" title="E51">A4</font></i></a>);<br/></i>
<a NAME="E6:69"/><i><font color="Green" title="E54">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:69_3_1"/><i><font color="Green" title="E55">A9</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:69_3_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_3.html#E4:69"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="group_3.html#E5:69"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="group_3.html#E3:69_3_1"><i><font color="Green" title="E55">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:69_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:69_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:69_3"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>;<br/>
<a NAME="E5:69_3"/><i><font color="Green" title="E56">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_3.html#E4:69_3"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="group_3.html#T30" target="_self" title="GROUP_3:th.30">Th30</a></i>;<br/>
<a NAME="E6:69_3"/><i><font color="Green" title="E57">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:69_3"/><i><font color="Green" title="E58">A13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_3.html#E5:69"><i><font color="Green" title="E53">A7</font></i></a></i>;<br/>
<a NAME="E9:69_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_3.html#E1:69"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="group_3.html#E6:69_3"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="group_3.html#E8:69_3"><i><font color="Green" title="E58">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:69_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="group_3.html#E4:69"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="group_3.html#E5:69_3"><i><font color="Green" title="E56">A11</font></i></a>, <a class="txt" href="group_3.html#E6:69_3"><i><font color="Green" title="E57">A12</font></i></a>, <a class="txt" href="group_3.html#E8:69_3"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:69"/>
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:69_4"/><i><font color="Green" title="E55">A14</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E2:69_4"/><i><font color="Green" title="E56">A15</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E3:69_4"/><i><font color="Green" title="E57">A16</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:69_4"/><i><font color="Green" title="E58">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font>
 <b>and </b><br/><a NAME="E6:69_4"/><i><font color="Green" title="E59">A18</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E4:69"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="group_3.html#E5:69"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="group_3.html#E1:69_4"><i><font color="Green" title="E55">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:69_4"/><i><font color="Green" title="E60">A19</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c3">H</font>
 <b>and </b><br/><a NAME="E9:69_4"/><i><font color="Green" title="E61">A20</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E4:69"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="group_3.html#E5:69"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="group_3.html#E2:69_4"><i><font color="Green" title="E56">A15</font></i></a></i>;<br/>
<a NAME="E10:69_4"/><i><font color="Green" title="E62">A21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E3:69_4"><i><font color="Green" title="E57">A16</font></i></a>, <a class="txt" href="group_3.html#E6:69_4"><i><font color="Green" title="E59">A18</font></i></a>, <a class="txt" href="group_3.html#E9:69_4"><i><font color="Green" title="E61">A20</font></i></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T32" target="_self" title="GROUP_3:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_3.html#T28" target="_self" title="GROUP_3:th.28">Th28</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E13:69_4"/><i><font color="Green" title="E63">A22</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="group_3.html#E10:69_4"><i><font color="Green" title="E62">A21</font></i></a>, <a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></i>;<br/>
<a NAME="E14:69_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E13:69_4"><i><font color="Green" title="E63">A22</font></i></a>, <a class="ref" href="group_3.html#T21" target="_self" title="GROUP_3:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E15:69_4"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="group_3.html#E5:69_4"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="group_3.html#E8:69_4"><i><font color="Green" title="E60">A19</font></i></a>, <a class="txt" href="group_3.html#E13:69_4"><i><font color="Green" title="E63">A22</font></i></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:69"/><b>then </b>
 <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">H</font>, <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="group_3.html#E4:69"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="group_3.html#E6:69"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:69"/>
 <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>


</div>
