<?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">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">g</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">g</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> );<br/>
<a NAME="E1:127"/><i><font color="Green" title="E89">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="c2">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="127_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:127_1"/>
<font color="Maroon" title="c3">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="c2">H</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c6">a</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="E2:127_1"/><i><font color="Green" title="E90">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E3:127_1"/><i><font color="Green" title="E91">A3</font></i>: 
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 ;<br/>

<b>given </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><a NAME="E4:127_1"/><i><font color="Green" title="E92">A4</font></i>: 
<font color="Maroon" title="c3">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="c2">H</font>
 <b>and </b><br/><a NAME="E5:127_1"/><i><font color="Green" title="E93">A5</font></i>: 
<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 ;<br/>

<a NAME="E6:127_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</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="c6">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_2.html#E2:127_1"><i><font color="Green" title="E90">A2</font></i></a>, <a class="txt" href="group_2.html#E4:127_1"><i><font color="Green" title="E92">A4</font></i></a>, <a class="ref" href="group_2.html#T137" target="_self" title="GROUP_2:th.137">Th137</a></i>;<br/>
<a NAME="E7:127_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:127_1"/>
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font>
 <b>by </b><i><a class="txt" href="group_2.html#E3:127_1"><i><font color="Green" title="E91">A3</font></i></a>, <a class="txt" href="group_2.html#E5:127_1"><i><font color="Green" title="E93">A5</font></i></a>, <a class="ref" href="group_2.html#T143" target="_self" title="GROUP_2:th.143">Th143</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:127"/><i><font color="Green" title="E90">A6</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="c2">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="127_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:127_2"/>
<font color="Maroon" title="c3">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="c2">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">g</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="E3:127_2"/><i><font color="Green" title="E91">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#D15" target="_self" title="GROUP_2:def.15">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c5">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c4">g</font>
;<br/>

<b>thus </b><a NAME="E5:127_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="group_2.html#E3:127_2"><i><font color="Green" title="E91">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:127"/><i><font color="Green" title="E91">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">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="c2">H</font>
 <b>and </b><br/><a NAME="E5:127"/><i><font color="Green" title="E92">A9</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="c2">H</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">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_2.html#E1:127"><i><font color="Green" title="E89">A1</font></i></a>, <a class="txt" href="group_2.html#E2:127"><i><font color="Green" title="E90">A6</font></i></a>);<br/></i>
<a NAME="E6:127"/><i><font color="Green" title="E93">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font>
  <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="127_3_1"><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> ; <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="E1:127_3_1"/>
<font color="Maroon" title="c4">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="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:127_3_1"/><i><font color="Green" title="E94">A11</font></i>: 
<font color="Maroon" title="c5">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="c3">f</font>
 <b>and </b><br/><a NAME="E4:127_3_1"/><i><font color="Green" title="E95">A12</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="E5:127_3_1"/>
 ex <font color="Olive" title="b1">g</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="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="group_2.html#E4:127"><i><font color="Green" title="E91">A8</font></i></a>, <a class="txt" href="group_2.html#E5:127"><i><font color="Green" title="E92">A9</font></i></a>, <a class="txt" href="group_2.html#E3:127_3_1"><i><font color="Green" title="E94">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:127_3_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="group_2.html#E4:127_3_1"><i><font color="Green" title="E95">A12</font></i></a>, <a class="ref" href="group_2.html#D16" target="_self" title="GROUP_2:def.16">Def16</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">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="E2:127_3"/><i><font color="Green" title="E94">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">g</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="E5:127_3"/><i><font color="Green" title="E95">A14</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="group_2.html#E2:127_3"><i><font color="Green" title="E94">A13</font></i></a>, <a class="ref" href="group_2.html#D16" target="_self" title="GROUP_2:def.16">Def16</a></i>;<br/>
<a NAME="E6:127_3"/><i><font color="Green" title="E96">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</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="c2">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="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D15" target="_self" title="GROUP_2:def.15">Def15</a></i>;<br/>
<a NAME="E7:127_3"/><b>then </b><i><font color="Green" title="E97">A16</font></i>: 
<font color="Maroon" title="c3">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">g</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="c2">H</font></span>)</span> <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="c3">f</font>
 
<b>by </b><i><a class="txt" href="group_2.html#E4:127"><i><font color="Green" title="E91">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a</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="E9:127_3"/><i><font color="Green" title="E98">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</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="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E10:127_3"/><i><font color="Green" title="E99">A18</font></i>: 
<font color="Maroon" title="c3">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">g</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="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E5:127"><i><font color="Green" title="E92">A9</font></i></a>, <a class="txt" href="group_2.html#E6:127_3"><i><font color="Green" title="E96">A15</font></i></a></i>;<br/>
<a NAME="E11:127_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_2.html#E9:127_3"><i><font color="Green" title="E98">A17</font></i></a>, <a class="ref" href="group_2.html#T137" target="_self" title="GROUP_2:th.137">Th137</a></i>;<br/>
<b>hence </b><a NAME="E12:127_3"/>
<font color="Maroon" title="c4">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="c3">f</font>
 <b>by </b><i><a class="txt" href="group_2.html#E5:127_3"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="group_2.html#E7:127_3"><i><font color="Green" title="E97">A16</font></i></a>, <a class="txt" href="group_2.html#E10:127_3"><i><font color="Green" title="E99">A18</font></i></a>, <a class="ref" href="group_2.html#T143" target="_self" title="GROUP_2:th.143">Th143</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:127"/>
<font color="Maroon" title="c3">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="127_4"><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> ; <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="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:127_4"/><i><font color="Green" title="E94">A19</font></i>: 
<font color="Maroon" title="c4">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="c3">f</font>
 <b>and </b><br/><a NAME="E2:127_4"/><i><font color="Green" title="E95">A20</font></i>: 
<font color="Maroon" title="c5">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="c3">f</font>
 <b>and </b><br/><a NAME="E3:127_4"/><i><font color="Green" title="E96">A21</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">a</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="E5:127_4"/><i><font color="Green" title="E97">A22</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E6:127_4"/><i><font color="Green" title="E98">A23</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E4:127"><i><font color="Green" title="E91">A8</font></i></a>, <a class="txt" href="group_2.html#E5:127"><i><font color="Green" title="E92">A9</font></i></a>, <a class="txt" href="group_2.html#E1:127_4"><i><font color="Green" title="E94">A19</font></i></a></i>;<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="E8:127_4"/><i><font color="Green" title="E99">A24</font></i>: 
<font color="Maroon" title="c5">y</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="c2">H</font>
 <b>and </b><br/><a NAME="E9:127_4"/><i><font color="Green" title="E100">A25</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E4:127"><i><font color="Green" title="E91">A8</font></i></a>, <a class="txt" href="group_2.html#E5:127"><i><font color="Green" title="E92">A9</font></i></a>, <a class="txt" href="group_2.html#E2:127_4"><i><font color="Green" title="E95">A20</font></i></a></i>;<br/>
<a NAME="E10:127_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_2.html#E3:127_4"><i><font color="Green" title="E96">A21</font></i></a>, <a class="txt" href="group_2.html#E6:127_4"><i><font color="Green" title="E98">A23</font></i></a>, <a class="txt" href="group_2.html#E9:127_4"><i><font color="Green" title="E100">A25</font></i></a>, <a class="ref" href="group_2.html#T143" target="_self" title="GROUP_2:th.143">Th143</a></i>;<br/>
<a NAME="E11:127_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</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="c6">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E12:127_4"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="group_2.html#E5:127_4"><i><font color="Green" title="E97">A22</font></i></a>, <a class="txt" href="group_2.html#E8:127_4"><i><font color="Green" title="E99">A24</font></i></a>, <a class="ref" href="group_2.html#T137" target="_self" title="GROUP_2:th.137">Th137</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:127"/>
 <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font>, <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="group_2.html#E4:127"><i><font color="Green" title="E91">A8</font></i></a>, <a class="txt" href="group_2.html#E6:127"><i><font color="Green" title="E93">A10</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>


</div>
