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

<b>let </b><font color="Maroon">T</font> be   <a href="topgrp_1.html#NM4">TopGroup</a>;<br/>

<b>assume </b><a NAME="E1:81"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">W</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="group_1.html#K3">"</a> </span>)</span>  ex <font color="Olive">A</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b</font> st <font color="Olive">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font>
 ;<br/>

<a NAME="E2:81"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">W</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> <a href="group_1.html#K3">"</a>   ex <font color="Olive">A</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> st <font color="Olive">A</font> <a href="group_2.html#K1">"</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <a href="group_1.html#K3">"</a> ;<br/>



<a NAME="E1:81_1"/>
<font color="Maroon">W</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">T</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">T</font>, <font color="Maroon">B</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font><b> such that </b><br/><a NAME="E3:81_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>let </b><font color="Maroon">x</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="E4:81_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="group_2.html#K1">"</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:81_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T5">GROUP_2:5</a></i>;<br/>
<a NAME="E7:81_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E8:81_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">T</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<a NAME="E9:81_1"/><b>then </b>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:81_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:81"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">W</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font>  ex <font color="Olive">A</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b</font> st <font color="Olive">A</font> <a href="group_2.html#K2">*</a> <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>;<br/>



<a NAME="E1:81_2"/>
<font color="Maroon">W</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font>, <font color="Maroon">B</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">b</font> <a href="group_1.html#K3">"</a> <b> such that </b><br/><a NAME="E3:81_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>consider </b><font color="Maroon">B1</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">b</font><b> such that </b><br/><a NAME="E5:81_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">B1</font> <a href="group_2.html#K1">"</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">B1</font>
;<br/>

<b>let </b><font color="Maroon">x</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="E6:81_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:81_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<a NAME="E9:81_2"/>
<font color="Maroon">h</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i>, <a class="ref" href="group_2.html#T5">GROUP_2:5</a></i>;<br/>
<a NAME="E10:81_2"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:81_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<b>hence </b><a NAME="E12:81_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:81"/>
<font color="Maroon">T</font> is   <a href="topgrp_1.html#NM5">TopologicalGroup</a>
 <b>by </b><i>, <a class="ref" href="topgrp_1.html#T4" target="_self">Th4</a>, </i>;<br/>


</div>
