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<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" title="TOPGRP_1:func.5">HomeoGroup</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E32">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">a</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">G</font> =  <a href="topgrp_1.html#K5" title="TOPGRP_1:func.5">HomeoGroup</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E2:56"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a>  is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_1"/>
<font color="Maroon" title="c2">f</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a>  <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D4" target="_self" title="TOPGRP_1:def.4">TOPGRP_1:def 4</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c2">f</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" title="TOPGRP_1:func.5">HomeoGroup</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="topgrp_1.html#D5" target="_self" title="TOPGRP_1:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>now </b></a><div class="add"><a NAME="E1:56_2"/><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="topgrp_1.html#E2:56"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="topgrp_1.html#D5" target="_self" title="TOPGRP_1:def.5">Def5</a></i>
<br/>.= 
 <a href="topgrp_1.html#K4" title="TOPGRP_1:func.4">id</a> <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:56_2"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" title="TOPGRP_1:func.5">HomeoGroup</a> <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#T33" target="_self" title="TOPGRP_1:th.33">Th33</a></i>
;<br/><b>thus </b><font color="Maroon" title="c5">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a> </span>)</span>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="topgrp_1.html#E2:56"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="topgrp_1.html#D5" target="_self" title="TOPGRP_1:def.5">Def5</a></i>
<br/>.= 
 <a href="topgrp_1.html#K4" title="TOPGRP_1:func.4">id</a> <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:56_2"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" title="TOPGRP_1:func.5">HomeoGroup</a> <font color="Maroon" title="c1">T</font></span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#T33" target="_self" title="TOPGRP_1:th.33">Th33</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:56"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="tops_2.html#K2" title="TOPS_2:func.2">/"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T12" title="GROUP_1:th.12">GROUP_1:12</a></i>;<br/>


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