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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>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">g</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">g</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K3">"</a> </span>)</span> );<br/>
<a NAME="E1:127"/><i><font color="Green">E35</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" target="_self">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="127_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>assume </b><a NAME="E1:127_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font>
 ;<br/>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:127_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E3:127_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 ;<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="E4:127_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:127_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
 ;<br/>

<a NAME="E6:127_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:127_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <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_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:127_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, , <a class="ref" href="group_2.html#T50" target="_self">Th50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:127"/><i><font color="Green">E51</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" target="_self">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="127_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="E1:127_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:127_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </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">g</font>
;<br/>

<b>thus </b><a NAME="E5:127_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </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="E4:127"/><i><font color="Green">E53</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" target="_self">Left_Cosets</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:127"/><i><font color="Green">E54</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" target="_self">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="E6:127"/><i><font color="Green">E55</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#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:127_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#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font>
  <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="127_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="E1:127_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="E3:127_3_1"/><i><font color="Green">E56</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="E4:127_3_1"/><i><font color="Green">E57</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> <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="E5:127_3_1"/>
 ex <font color="Olive">g</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">g</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</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">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K3">"</a> </span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:127_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font>
 <b>by </b><i>, </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="E2:127_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:127_3"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:127_3"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:127_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<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">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font></span>)</span> <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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></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><b> such that </b><br/><a NAME="E9:127_3"/><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E10:127_3"/><i><font color="Green">E135</font></i>: 
<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">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="group_2.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E11:127_3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <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">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55" target="_self">Th55</a>, <a class="ref" href="group_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E12:127_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="ref" href="group_2.html#T57" target="_self">Th57</a>, <a class="ref" href="group_2.html#T50" target="_self">Th50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:127"/>
<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="127_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="E1:127_4"/><i><font color="Green">E136</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="E2:127_4"/><i><font color="Green">E137</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="E3:127_4"/><i><font color="Green">E138</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">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:127_4"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:127_4"/><i><font color="Green">E179</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">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, , </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><b> such that </b><br/><a NAME="E8:127_4"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E9:127_4"/><i><font color="Green">E181</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> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E10:127_4"/>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <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_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T53" target="_self">Th53</a>, <a class="ref" href="group_2.html#T55" target="_self">Th55</a>, <a class="ref" href="group_2.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E11:127_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E12:127_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_2.html#T57" target="_self">Th57</a>, <a class="ref" href="group_2.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:127"/>
 <a href="group_2.html#K14" target="_self">Left_Cosets</a> <font color="Maroon">H</font>, <a href="group_2.html#K15" target="_self">Right_Cosets</a> <font color="Maroon">H</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i>, , <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>


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