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

<b>reconsider </b><font color="Maroon" title="c2">K1</font> =  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">K2</font> =  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">K1</font> <b>by </b><i><a class="ref" href="autgroup.html#T14" target="_self" title="AUTGROUP:th.14">Th14</a></i>;<br/>
<a NAME="E3:23_1_1"/>
for <font color="Olive" title="b1">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">K2</font>,<font color="Maroon" title="c3">K2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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" title="c4">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">K2</font>,<font color="Maroon" title="c3">K2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K2</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K2</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="autgroup.html#E3:23_1_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<a NAME="E5:23_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">y</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#T18" target="_self" title="AUTGROUP:th.18">Th18</a></i>;<br/>
<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> = 
<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="txt" href="autgroup.html#E3:23_1_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">y</font>
<b>by </b><i><a class="ref" href="autgroup.html#T15" target="_self" title="AUTGROUP:th.15">Th15</a></i>
;<br/>
<b>hence </b><a NAME="E7:23_1_1_1"/>
<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K2</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E5:23_1_1_1"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:23_1_1"/><b>then </b>
 <a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font> is    <a href="realset1.html#M2" title="REALSET1:mode.2">Presv</a> of <font color="Maroon" title="c2">K1</font>,<font color="Maroon" title="c3">K2</font>
 
<b>by </b><i><a class="ref" href="realset1.html#D6" title="REALSET1:def.6">REALSET1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">op</font> = <span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="realset1.html#T9" title="REALSET1:th.9">REALSET1:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">IG</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #);<br/>
<a NAME="E6:23_1_1"/>
(  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> &amp;  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">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" title="E21">A3</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" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #);<br/><b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:23_1_1_2_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> )
 ;<br/><a NAME="E2:23_1_1_2_1"/><i><font color="Green" title="E23">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y</font> = 
<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="autgroup.html#E2:23_1_1_2_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="ref" href="autgroup.html#T15" target="_self" title="AUTGROUP:th.15">Th15</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:23_1_1_2"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">h</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">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" title="c6">f</font>, <font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c8">h</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E2:23_1_1_2_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>
<a NAME="E3:23_1_1_2_2"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">C</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">C</font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="autgroup.html#E2:23_1_1_2_2"><i><font color="Green" title="E22">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">C</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">h</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="autgroup.html#E3:23_1_1_2_2"><i><font color="Green" title="E23">A7</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E3:23_1_1_2"/>
 ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) st <br/>for <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) holds <br/> ( <font color="Olive" title="b2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp; <font color="Olive" title="b1">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp;  ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) st <br/>( <font color="Olive" title="b2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> ) )
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">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" title="c6">e</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) <b>by </b><i><a class="ref" href="autgroup.html#T16" target="_self" title="AUTGROUP:th.16">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">e</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #);<br/>

<b>consider </b><font color="Maroon" title="c8">A</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:23_1_1_2_3"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 ;<br/>
<font color="Maroon" title="c7">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">e</font> = 
<font color="Maroon" title="c8">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="autgroup.html#E3:23_1_1_2_3"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E5:23_1_1_2_3"/>
<font color="Maroon" title="c7">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:23_1_1_2_3"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>

<font color="Maroon" title="c6">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="autgroup.html#E3:23_1_1_2_3"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:23_1_1_2_3"/>
<font color="Maroon" title="c6">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:23_1_1_2_3"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) <b>by </b><i><a class="ref" href="autgroup.html#T17" target="_self" title="AUTGROUP:th.17">Th17</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">g</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c8">A</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#T13" target="_self" title="AUTGROUP:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A</font> = <font color="Maroon" title="c10">A</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<a NAME="E11:23_1_1_2_3"/><i><font color="Green" title="E23">A9</font></i>: 
( <font color="Maroon" title="c11">A</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c11">A</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  )
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<a NAME="E12:23_1_1_2_3"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_6.html#D13" title="GROUP_6:def.13">GROUP_6:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c7">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">g</font> = 
<font color="Maroon" title="c11">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="autgroup.html#E3:23_1_1_2_3"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="txt" href="autgroup.html#E11:23_1_1_2_3"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="autgroup.html#E12:23_1_1_2_3"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c9">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">A</font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="autgroup.html#E3:23_1_1_2_3"><i><font color="Green" title="E22">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="txt" href="autgroup.html#E11:23_1_1_2_3"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="autgroup.html#E12:23_1_1_2_3"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">IG</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c4">op</font> #) as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<a NAME="E8:23_1_1"/>
<font color="Maroon" title="c6">IG</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">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" title="E21">A11</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">IG</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="autgroup.html#T14" target="_self" title="AUTGROUP:th.14">Th14</a></i>;<br/>
<b>thus </b><a NAME="E2:23_1_1_3"/>
<font color="Maroon" title="c6">IG</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_3"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">IG</font> = <font color="Maroon" title="c6">IG</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E10:23_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">IG</font> holds <br/><font color="Olive" title="b2">k</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Olive" title="b1">f</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">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" title="c8">f</font>, <font color="Maroon" title="c9">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_1_1_4"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c9">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">IG</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">f1</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:23_1_1_4"/><i><font color="Green" title="E22">A13</font></i>: 
<font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">g</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:23_1_1_4"/><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1:23_1_1_4"><i><font color="Green" title="E21">A12</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_4"/>
<font color="Maroon" title="c11">g</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#T13" target="_self" title="AUTGROUP:th.13">Th13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">g1</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:23_1_1_4"/><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c12">g1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">g</font>
 ;<br/>
<a NAME="E9:23_1_1_4"/>
<font color="Maroon" title="c12">g1</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">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="E11:23_1_1_4"/><i><font color="Green" title="E25">A16</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c12">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c13">a</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E8:23_1_1_4"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="autgroup.html#D4" target="_self" title="AUTGROUP:def.4">Def4</a></i>;<br/>
<a NAME="E12:23_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">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" title="c10">f1</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<a NAME="E2:23_1_1_4_1"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
<font color="Maroon" title="c10">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<i><font color="Green" title="E27">A18</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f1</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font>
<b>by </b><i><a class="txt" href="autgroup.html#E9"><i><font color="Green" title="E6">Lm3</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="txt" href="autgroup.html#E9"><i><font color="Green" title="E6">Lm3</font></i></a></i>
;<br/>
<a NAME="E4:23_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E2:23_1_1_4_1"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">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" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g1</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">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" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/>
<a NAME="E7:23_1_1_4_1"/><b>then </b><i><font color="Green" title="E28">A19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E8:23_1_1_4_1"/>
<font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#T7" target="_self" title="AUTGROUP:th.7">Th7</a></i>;<br/>
<a NAME="E9:23_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">A</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E11:23_1_1_4_1"/><i><font color="Green" title="E29">A20</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 ;<br/>
<a NAME="E12:23_1_1_4_1"/><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E2:23_1_1_4_1"><i><font color="Green" title="E26">A17</font></i></a>, <a class="txt" href="autgroup.html#E3:23_1_1_4_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="autgroup.html#E11:23_1_1_4_1"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">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" title="c15">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <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="c12">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <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="c10">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E11:23_1_1_4"><i><font color="Green" title="E25">A16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">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="p3">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E11:23_1_1_4_1"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E12:23_1_1_4_1"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c15">y</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T41" title="GROUP_6:th.41">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" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E7:23_1_1_4_1"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="autgroup.html#D4" target="_self" title="AUTGROUP:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:23_1_1_4"/><b>then </b><i><font color="Green" title="E26">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E8:23_1_1_4"><i><font color="Green" title="E24">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">C</font> = <font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#T7" target="_self" title="AUTGROUP:th.7">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">D</font> = <font color="Maroon" title="c11">g</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#T13" target="_self" title="AUTGROUP:th.13">Th13</a></i>;<br/>
<a NAME="E16:23_1_1_4"/>
<font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 
<b>by </b><i><a class="txt" href="autgroup.html#E3:23_1_1_4"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="autgroup.html#T11" target="_self" title="AUTGROUP:th.11">Th11</a></i>;<br/>
<a NAME="E17:23_1_1_4"/><b>then </b><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon" title="c14">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E5:23_1_1_4"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="autgroup.html#T9" target="_self" title="AUTGROUP:th.9">Th9</a></i>;<br/>
<a NAME="E18:23_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">g</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">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" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c11">g</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="autgroup.html#T7" target="_self" title="AUTGROUP:th.7">Th7</a>, <a class="ref" href="autgroup.html#T13" target="_self" title="AUTGROUP:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E2:23_1_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">g</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="autgroup.html#T8" target="_self" title="AUTGROUP:th.8">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">E</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">g</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> ;<br/>
<b>consider </b><font color="Maroon" title="c17">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:23_1_1_4"/><i><font color="Green" title="E28">A24</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">f</font>
 ;<br/>
<a NAME="E22:23_1_1_4"/>
<font color="Maroon" title="c16">E</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E3:23_1_1_4"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="autgroup.html#E17:23_1_1_4"><i><font color="Green" title="E27">A23</font></i></a>, <a class="ref" href="autgroup.html#T9" target="_self" title="AUTGROUP:th.9">Th9</a></i>;<br/>
<a NAME="E23:23_1_1_4"/><b>then </b>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c7">IG</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E13:23_1_1_4"><i><font color="Green" title="E26">A22</font></i></a>, <a class="txt" href="autgroup.html#E21:23_1_1_4"><i><font color="Green" title="E28">A24</font></i></a></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" title="c8">f</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">f</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">IG</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E21:23_1_1_4"><i><font color="Green" title="E28">A24</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:23_1_1_4"/>
<font color="Maroon" title="c9">k</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c8">f</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">IG</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">IG</font> = <font color="Maroon" title="c7">IG</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">IG</font>
;<br/>

<b>thus </b><a NAME="E12:23_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">IG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="autgroup.html#K4" title="AUTGROUP:func.4">InnAut</a> <font color="Maroon" title="c1">G</font>
 ;<br/>


</div>
