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<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/><b>let </b><font color="Maroon" title="c2">f</font> be    <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>let </b><font color="Maroon" title="c3">g</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="E1:14"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">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="E3:14"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 ;<br/>
<i><font color="Green" title="E15">A3</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</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="E5:14"/>
<font color="Maroon" title="c2">f</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="E6:14"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c2">f</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/>
<a NAME="E7:14"/>
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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="c3">g</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="autgroup.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</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#K3" title="AUTGROUP:func.3">AutGroup</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="E9:14"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</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="autgroup.html#T10" target="_self" title="AUTGROUP:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon" title="c2">f</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="c3">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="autgroup.html#E3:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="autgroup.html#E4:14"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="autgroup.html#E6:14"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="funct_2.html#T36" title="FUNCT_2:th.36">FUNCT_2:36</a></i>;<br/>


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