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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="topgrp_1.html#NM4" title="TOPGRP_1:NM.4">TopGroup</a>;<br/>

<b>assume </b><a NAME="E1:78"/><i><font color="Green" title="E40">A1</font></i>: 
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">T</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>   ex <font color="Olive" title="b3">A</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> st <font color="Olive" title="b3">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">f</font> =  <a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E2:78"/>
for <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">W</font>  ex <font color="Olive" title="b3">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">W</font> st <span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>;<br/>



<a NAME="E1:78_1"/>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#D7" title="GROUP_1:def.7">GROUP_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">A</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">a</font><b> such that </b><br/><a NAME="E3:78_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E1:78"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">A</font>
;<br/>

<b>thus </b><a NAME="E4:78_1"/>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E3:78_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="topgrp_1.html#T10" target="_self" title="TOPGRP_1:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:78"/>
 <a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D3" title="BORSUK_1:def.3">BORSUK_1:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D7" target="_self" title="TOPGRP_1:def.7">TOPGRP_1:def 7</a><br/>


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