<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a> ;<br/>
<a NAME="E1:82_1_1"/>
for <font color="Olive">w</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font><br/> for <font color="Olive">A</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font>  ex <font color="Olive">W</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">w</font> st <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>;<br/>



<a NAME="E1:82_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">B</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font>, <font color="Maroon">W</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">w</font><b> such that </b><br/><a NAME="E3:82_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">B</font> <a href="group_2.html#K2">*</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:82_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E5:82_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:82_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T33">GROUP_2:33</a></i>;<br/>
<a NAME="E8:82_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E9:82_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="group_2.html#K2">*</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<b>hence </b><a NAME="E10:82_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:82_1_1"/>
<font color="Maroon">a</font> <a href="topgrp_1.html#K1" target="_self">*</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>


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