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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<a NAME="E1:36"/><i><font color="Green" title="E28">A1</font></i>: 
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#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span> <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>
 
<b>by </b><i><a class="ref" href="autgroup.html#D5" target="_self" title="AUTGROUP:def.5">Def5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>) <b>-&gt; </b>   <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> =  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default">$1 <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <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>, <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="E3:36"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">a</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="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font> as   <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 <span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span> <b>by </b><i><a class="txt" href="autgroup.html#E1:36"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><i><font color="Green" title="E30">A3</font></i>: <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E3:36"><i><font color="Green" title="E29">A2</font></i></a></i>
<br/>.= 
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T25" title="GROUP_1:th.25">GROUP_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="autgroup.html#T23" target="_self" title="AUTGROUP:th.23">Th23</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</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#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/><b>assume </b><a NAME="E1:36_1_2"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> )
 ;<br/><a NAME="E2:36_1_2"/><b>then </b>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="autgroup.html#E3:36"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</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="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="autgroup.html#E1:36_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="autgroup.html#E1:36_1_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="autgroup.html#T20" target="_self" title="AUTGROUP:th.20">Th20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</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="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">f1</font> = <font color="Maroon" title="c3">f</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</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/>
<a NAME="E7:36"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c5">KER</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">b</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="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> </span>}</span> ;<br/>
<a NAME="E1:36_2"/><i><font color="Green" title="E31">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_6.html#D10" title="GROUP_6:def.10">GROUP_6:def 10</a></i>;<br/>
<a NAME="E2:36_2"/><i><font color="Green" title="E32">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">b</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="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_2_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 ;<br/>

<a NAME="E2:36_2_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="autgroup.html#T21" target="_self" title="AUTGROUP:th.21">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</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="E4:36_2_1"/><i><font color="Green" title="E34">A9</font></i>: 
( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#E1:36_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="autgroup.html#E1:36_2_1"><i><font color="Green" title="E33">A8</font></i></a></i>;<br/>
<a NAME="E5:36_2_1"/><i><font color="Green" title="E35">A10</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<a NAME="E1:36_2_1_1"/>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:36_2_1_1"/>
<span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:36"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="autgroup.html#E4:36_2_1"><i><font color="Green" title="E34">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36_2_1"/><i><font color="Green" title="E36">A11</font></i>: 
for <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<a NAME="E1:36_2_1_2"/>
<span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="autgroup.html#D6" target="_self" title="AUTGROUP:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E2:36_2_1_2"/>
<font color="Maroon" title="c9">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E5:36_2_1"><i><font color="Green" title="E35">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:36_2_1"/>
for <font color="Olive" title="b1">b</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="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<a NAME="E1:36_2_1_3"/>
<font color="Maroon" title="c9">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</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="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>
<a NAME="E2:36_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E6:36_2_1"><i><font color="Green" title="E36">A11</font></i></a></i>;<br/>
<a NAME="E3:36_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</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="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E4:36_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:36_2_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:36_2_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">b</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="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="autgroup.html#E4:36_2_1"><i><font color="Green" title="E34">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">b</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="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> </span>}</span>  <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="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_2_2"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">b</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="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:36_2_2"/><i><font color="Green" title="E33">A12</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; ( for <font color="Olive" title="b1">b</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="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</font> ) )
 ;<br/>
<a NAME="E4:36_2_2"/><i><font color="Green" title="E34">A13</font></i>: 
for <font color="Olive" title="b1">b</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="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<a NAME="E1:36_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</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="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">x</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="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="autgroup.html#E3:36_2_2"><i><font color="Green" title="E33">A12</font></i></a></i>;<br/>
<a NAME="E2:36_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</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="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E3:36_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</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="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E4:36_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</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="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E5:36_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E6:36_2_2_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="autgroup.html#D6" target="_self" title="AUTGROUP:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">g</font> being   <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> such that </b><br/><a NAME="E6:36_2_2"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 ;<br/>
<a NAME="E7:36_2_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<a NAME="E1:36_2_2_2"/>
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> )
 
<b>by </b><i><a class="txt" href="autgroup.html#E4:36_2_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="autgroup.html#E6:36_2_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:36_2_2_2"/>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:36_2_2"/><b>then </b>
<font color="Maroon" title="c9">g</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="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E9:36_2_2"/><b>then </b>
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="autgroup.html#E6:36_2_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="autgroup.html#T21" target="_self" title="AUTGROUP:th.21">Th21</a></i>;<br/>
<a NAME="E10:36_2_2"/><b>then </b>
<font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="autgroup.html#E3:36"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:36_2_2"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="autgroup.html#E1:36_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="autgroup.html#E3:36_2_2"><i><font color="Green" title="E33">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:36_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">b</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="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="autgroup.html#E2:36_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:36_2"/>
 <a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c4">f1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_5.html#D10" title="GROUP_5:def.10">GROUP_5:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:36"/>
<font color="Maroon" title="c4">f1</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<a NAME="E1:36_3"/>
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> 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#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">q1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_3_1"/><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ;<br/>

<a NAME="E2:36_3_1"/><b>then </b><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c5">q</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#D5" target="_self" title="AUTGROUP:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y1</font> being   <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> such that </b><br/><a NAME="E4:36_3_1"/><i><font color="Green" title="E33">A17</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 ;<br/>
<a NAME="E5:36_3_1"/>
<font color="Maroon" title="c6">y1</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#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:36_3_1"/><i><font color="Green" title="E34">A18</font></i>: 
for <font color="Olive" title="b1">a</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="c6">y1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1:36"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="autgroup.html#E1:36_3_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="autgroup.html#E4:36_3_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="autgroup.html#D4" target="_self" title="AUTGROUP:def.4">Def4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c7">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> ;<br/>

<b>thus </b><a NAME="E8:36_3_1"/>
<font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">q1</font> = 
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E3:36"><i><font color="Green" title="E29">A2</font></i></a></i>
<br/>.= 
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c7">b</font>
<b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>
;<br/>
<b>hence </b><a NAME="E10:36_3_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">q1</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E2:36_3_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="autgroup.html#E4:36_3_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="txt" href="autgroup.html#E7:36_3_1"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="autgroup.html#D6" target="_self" title="AUTGROUP:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f1</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 <span class="p1">(<span class="default"><a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a></i>;<br/>
<b>hence </b><a NAME="E3:36_3"/>
<font color="Maroon" title="c4">f1</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D13" title="GROUP_6:def.13">GROUP_6:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:36"/><b>then </b>
 <a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c4">f1</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T67" title="GROUP_6:th.67">GROUP_6:67</a></i>;<br/>
<b>hence </b><a NAME="E10:36"/>
 <a href="autgroup.html#K5" title="AUTGROUP:func.5">InnAutGroup</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <span class="p1">(<span class="default"><a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="autgroup.html#E7:36"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="group_6.html#T90" title="GROUP_6:th.90">GROUP_6:90</a></i>;<br/>


</div>
