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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="topgrp_1.html#NM5" title="TOPGRP_1:NM.5">TopologicalGroup</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>;<br/>





<b>consider </b><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E2:103"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T41" target="_self" title="TOPGRP_1:th.41">Th41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font> <b>by </b><i><a class="ref" href="connsp_2.html#T4" title="CONNSP_2:th.4">CONNSP_2:4</a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">B</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:103"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:103"/><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )
 <b>by </b><i><a class="ref" href="group_2.html#T12" title="GROUP_2:th.12">GROUP_2:12</a></i>;<br/>
<a NAME="E7:103"/>
<font color="Maroon" title="c9">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E6:103"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="topgrp_1.html#T7" target="_self" title="TOPGRP_1:th.7">Th7</a></i>;<br/>
<a NAME="E8:103"/><b>then </b>
<font color="Maroon" title="c9">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:103"/><b>then </b>
( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E6:103"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="topgrp_1.html#T7" target="_self" title="TOPGRP_1:th.7">Th7</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:103"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E6:103"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="group_2.html#T12" title="GROUP_2:th.12">GROUP_2:12</a></i>;<br/>
<b>hence </b><a NAME="E11:103"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:103"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>


</div>
