<?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#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<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">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> );<br/>
<a NAME="E1:69"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">A</font> =  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>;<br/>
<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font>
 ;<br/>

<b>given </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:69_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>

<b>given </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E3:69_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/>

<b>thus </b><font color="Maroon">y1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_3.html#T5" target="_self">Th5</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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T39">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p4">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T43">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p4">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p3">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T38">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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T39">GROUP_2:39</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="group_3.html#T5" target="_self">Th5</a>, </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:69"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  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="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:69_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</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">y</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

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

<b>thus </b><a NAME="E5:69_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:69"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:69"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E6:69"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">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">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_3_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">y</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:69_3_1"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_3.html#E5:69_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_3.html#T19" target="_self">Th19</a>, <a class="txt" href="group_3.html#E6:69_3"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:69_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:69_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:69_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">c</font> = <font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>;<br/>
<a NAME="E5:69_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_3.html#E6:69_3"><i><font color="Green">E59</font></i></a>, </i>;<br/>
<a NAME="E6:69_3"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:69_3"/><i><font color="Green">E60</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="group_3.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E9:69_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="group_3.html#T20" target="_self">Th20</a>, <a class="ref" href="group_3.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E10:69_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="group_3.html#E5:69_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="group_3.html#E8:69_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="group_3.html#T20" target="_self">Th20</a>, <a class="ref" href="group_3.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:69"/>
<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="69_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:69_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:69_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:69_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:69_4"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:69_4"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="group_3.html#T19" target="_self">Th19</a>, </i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:69_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E9:69_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="group_3.html#T19" target="_self">Th19</a>, </i>;<br/>
<a NAME="E10:69_4"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T20" target="_self">Th20</a>, <a class="ref" href="group_3.html#T22" target="_self">Th22</a>, <a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_2.html#T137">GROUP_2:137</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E13:69_4"/><i><font color="Green">E116</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font> &amp; <font color="Maroon">d</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T25" target="_self">Th25</a>, <a class="ref" href="group_3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E14:69_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T26" target="_self">Th26</a>, </i>;<br/>
<b>hence </b><a NAME="E15:69_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_3.html#T21" target="_self">Th21</a>, <a class="ref" href="group_3.html#T23" target="_self">Th23</a>, <a class="ref" href="group_3.html#T26" target="_self">Th26</a>, <a class="ref" href="group_2.html#T137">GROUP_2:137</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:69"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font>, <a href="group_2.html#K14">Left_Cosets</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, , <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>thus </b><a NAME="E9:69"/>
 <a href="group_2.html#K16">Index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K16">Index</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
