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

<b>let </b><font color="Maroon">G</font> be   <a href="topgrp_1.html#NM5">TopologicalGroup</a>;<br/>

<a NAME="E1:110"/>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> st <br/>for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/> ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">e</font> =  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>;<br/>
<b>take </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:110_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:110_1"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E3:110_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:110_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>, <font color="Maroon">B</font> being  <a href="pre_topc.html#V3">open</a>  <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">G</font></span>)</span> <a href="group_1.html#K3">"</a> <b> such that </b><br/><a NAME="E6:110_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">C</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:110_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<a NAME="E8:110_1"/><b>then </b>
<font color="Maroon">B</font> <a href="group_2.html#K1">"</a>  is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E15:110_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span> as    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="connsp_2.html#T4">CONNSP_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<a NAME="E10:110_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E11:110_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>hence </b><a NAME="E12:110_1"/><i><font color="Green">E49</font></i>: 
(  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">p</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="E13:110_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">W</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E15:110_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="group_2.html#K3">*</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="topgrp_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E16:110_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E17:110_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="group_2.html#K3">*</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="E18:110_1"/><b>then </b>
<font color="Maroon">r</font> <a href="group_2.html#K3">*</a> <font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:110_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="group_2.html#K3">*</a> <font color="Maroon">W</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E21:110_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="group_2.html#K3">*</a> <font color="Maroon">W</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="topgrp_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E24:110_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="topgrp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="group_2.html#T33">GROUP_2:33</a></i>;<br/>
<a NAME="E25:110_1"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:110_1"/><b>then </b>
<font color="Maroon">W</font> <a href="group_2.html#K1">"</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E27:110_1"/><b>then </b>
<font color="Maroon">W</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/>
<a NAME="E28:110_1"/><b>then </b>
<font color="Maroon">C</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E29:110_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">C</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green">E85</font></i>: <font color="Maroon">r</font> = 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/>
<a NAME="E31:110_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E32:110_1"/>
<font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="group_2.html#T5">GROUP_2:5</a></i>;<br/>
<a NAME="E33:110_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T14" target="_self">Th14</a>, , <a class="ref" href="topgrp_1.html#D2" target="_self">Def2</a>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<b>hence </b><a NAME="E34:110_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:110"/>
<font color="Maroon">G</font> <a href="compts_1.html#NR3" target="_self">is_T3</a> 
 <b>by </b><i/>;<br/>


</div>
