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<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>;<br/>



<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:21"/>
<font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 
;<br/>
<b>then consider </b><font color="Maroon">f1</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 ;<br/>
<a NAME="E5:21"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E7:21"/>
<font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 
;<br/>
<b>then consider </b><font color="Maroon">f2</font> being   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:21"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 ;<br/>
<a NAME="E10:21"/>
( <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f2</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 
<b>by </b><i>, <a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/>
<a NAME="E11:21"/><b>then </b>
( <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="group_6.html#D13">GROUP_6:def 13</a></i>;<br/>
<a NAME="E12:21"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E13:21"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>
<a NAME="E14:21"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_3.html#K2">|^</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:21_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">a</font> = <font color="Maroon">b</font> <a href="group_1.html#K3">"</a> ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<a NAME="E3:21_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f1</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E5:21_1"/>
<font color="Maroon">f1</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 
<b>by </b><i><a class="txt" href="autgroup.html#E9:21"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<a NAME="E6:21_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f1</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  &amp; <font color="Maroon">f1</font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a>  )
 
<b>by </b><i><a class="ref" href="group_6.html#D14">GROUP_6:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:21_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="group_6.html#T68">GROUP_6:68</a></i>;<br/>
<a NAME="E9:21_1"/>
<font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="group_6.html#D12">GROUP_6:def 12</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">y1</font>
<b>by </b><i><a class="txt" href="autgroup.html#E2"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">y1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E5:21"><i><font color="Green">E24</font></i></a>, <a class="txt" href="autgroup.html#E2"><i><font color="Green">Lemma27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
;<br/>
<b>hence </b><a NAME="E11:21_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:21"/>
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K4" target="_self">InnAut</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E4:21"><i><font color="Green">E23</font></i></a>, </i>;<br/>


</div>
