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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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   <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <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> <font color="Olive" title="b1">a</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <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> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span><br/>



<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c2">a</font>;<br/>
<b>set </b><font color="Maroon" title="c4">g</font> =  <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>;<br/>
<a NAME="E1:33"/><i><font color="Green" title="E28">A1</font></i>: 
<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="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</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> <font color="Maroon" title="c2">a</font></span>)</span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>thus </b><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="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</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> <font color="Maroon" title="c2">a</font></span>)</span> = 
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="autgroup.html#T27" target="_self" title="AUTGROUP:th.27">Th27</a></i>
<br/>.= 
 <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#T24" target="_self" title="AUTGROUP:th.24">Th24</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:33"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c2">a</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 </b><i><font color="Green" title="E30">A3</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c2">a</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c2">a</font></span>)</span>
<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#E2:33"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="autgroup.html#E9"><i><font color="Green" title="E6">Lm3</font></i></a></i>
;<br/>
<a NAME="E4:33"/>
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c2">a</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="txt" href="autgroup.html#E2:33"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<a NAME="E5:33"/><b>then </b>
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <font color="Maroon" title="c2">a</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="autgroup.html#E2:33"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:33"/>
 <a href="autgroup.html#K6" title="AUTGROUP:func.6">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <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> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="autgroup.html#E1:33"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="autgroup.html#E3:33"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="funct_2.html#T36" title="FUNCT_2:th.36">FUNCT_2:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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