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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font>;<br/>
<a NAME="E2:57"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:57_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D4" target="_self">TOPGRP_1:def 4</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="txt" href="topgrp_1.html#E1:57"><i><font color="Green">E30</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><a NAME="E1:57_2"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i>, , <a class="txt" href="topgrp_1.html#E1:57"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
 <a href="topgrp_1.html#K4" target="_self">id</a> <font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> </span>)</span>
<b>by </b><i>, , <a class="txt" href="topgrp_1.html#E1:57"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
 <a href="topgrp_1.html#K4" target="_self">id</a> <font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
<font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="tops_2.html#K2">/"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T12">GROUP_1:12</a></i>;<br/>


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