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

<b>reconsider </b><font color="Maroon">K1</font> =  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">K2</font> =  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">K1</font> <b>by </b><i/>;<br/>
<a NAME="E3:23_1_1"/>
for <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">K2</font>,<font color="Maroon">K2</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">K2</font>,<font color="Maroon">K2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E5:23_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">y</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E7:23_1_1_1"/>
<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:23_1_1"/><b>then </b>
 <a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font> is    <a href="realset1.html#M2">Presv</a> of <font color="Maroon">K1</font>,<font color="Maroon">K2</font>
 
<b>by </b><i><a class="ref" href="realset1.html#D6">REALSET1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">op</font> = <span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font></span>)</span> <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span> as   <a href="binop_1.html#NM2">BinOp</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="realset1.html#T9">REALSET1:9</a></i>;<br/>
<b>set </b><font color="Maroon">IG</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #);<br/>
<a NAME="E6:23_1_1"/>
(  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) is <a href="group_1.html#V4">associative</a> &amp;  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) is <a href="group_1.html#V3">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #);<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>, <font color="Maroon">g</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:23_1_1_2_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/><a NAME="E2:23_1_1_2_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">G</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
<b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:23_1_1_2"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #), <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #), <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">f</font>, <font color="Maroon">B</font> = <font color="Maroon">g</font>, <font color="Maroon">C</font> = <font color="Maroon">h</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font> ;<br/>
<a NAME="E2:23_1_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:23_1_1_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">C</font>
<b>by </b><i>, <a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


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

<b>thus </b><a NAME="E3:23_1_1_2"/>
 ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) st <br/>for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Olive">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) )
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #);<br/>

<b>consider </b><font color="Maroon">A</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:23_1_1_2_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span>
<b>by </b><i>, <a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E5:23_1_1_2_3"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>

<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:23_1_1_2_3"/>
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">A</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E11:23_1_1_2_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">A</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E12:23_1_1_2_3"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_6.html#D13">GROUP_6:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>
;<br/>

<b>thus </b><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IG</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">op</font> #) as   <a href="group_1.html#NM1">Group</a> ;<br/>
<a NAME="E8:23_1_1"/>
<font color="Maroon">IG</font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:23_1_1_3"/><i><font color="Green">E81</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IG</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:23_1_1_3"/>
<font color="Maroon">IG</font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IG</font> = <font color="Maroon">IG</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of  <a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font> ;<br/>
<a NAME="E10:23_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font></span>)</span>  st <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IG</font> holds <br/><font color="Olive">k</font> <a href="group_3.html#K2">|^</a> <font color="Olive">f</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">IG</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font></span>)</span>, <font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_1_1_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IG</font>
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:23_1_1_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:23_1_1_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:23_1_1_4"/>
<font color="Maroon">g</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:23_1_1_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">g1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 ;<br/>
<a NAME="E9:23_1_1_4"/>
<font color="Maroon">g1</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then 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="E11:23_1_1_4"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:23_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:23_1_1_4_1"/>
<font color="Maroon">f1</font> is   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:23_1_1_4_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E89</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
<b>by </b><i/>
;<br/>
<a NAME="E4:23_1_1_4_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E5:23_1_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>
<a NAME="E6:23_1_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>
<a NAME="E7:23_1_1_4_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E8:23_1_1_4_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:23_1_1_4_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  is   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">A</font> being   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E11:23_1_1_4_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> 
 ;<br/>
<a NAME="E12:23_1_1_4_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="txt" href="autgroup.html#E2"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_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_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</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_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</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_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</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_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="group_3.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T41">GROUP_6:41</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:23_1_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:23_1_1_4"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">g</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E16:23_1_1_4"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/>
<a NAME="E17:23_1_1_4"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:23_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:23_1_1_4_2"/>
( <font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:23_1_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">E</font> = <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:23_1_1_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font>
 ;<br/>
<a NAME="E22:23_1_1_4"/>
<font color="Maroon">E</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="autgroup.html#E2"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<a NAME="E23:23_1_1_4"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">IG</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E24:23_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">IG</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:23_1_1_4"/>
<font color="Maroon">k</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">f</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">IG</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IG</font> = <font color="Maroon">IG</font> as  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of  <a href="autgroup.html#K3" target="_self">AutGroup</a> <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">IG</font>
;<br/>

<b>thus </b><a NAME="E12:23_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IG</font> <a href="hidden.html#R1">=</a>  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>
 ;<br/>


</div>
