<?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/><b>let </b><font color="Maroon">B</font> be    <a href="yellow_8.html#M1">Basis</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">M</font> be  <a href="tops_1.html#V1">dense</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>





<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span> ;<br/>
<a NAME="E1:109"/><i><font color="Green">E30</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:109_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:109_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="group_2.html#K4">*</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:109_1"/>
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<b>thus </b><a NAME="E5:109_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:109"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:109_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:109_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="group_2.html#K4">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E5:109_2"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E6:109_2"/><b>then </b>
<font color="Maroon">V</font> <a href="group_2.html#K4">*</a> <font color="Maroon">x</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E2:109"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:109_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:109"/>
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> holds <br/>for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> holds <br/> ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span>  &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:109_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:109_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:109_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T11" target="_self">Th11</a>, <a class="txt" href="topgrp_1.html#E2:109"><i><font color="Green">E47</font></i></a></i>;<br/>
<i><font color="Green">E54</font></i>: <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> = 
<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"><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#T16">GROUP_1:16</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E5:109_3"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <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#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E6:109_3"/><b>then </b>
 <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> <a href="group_2.html#K4">*</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="topgrp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<a NAME="E7:109_3"/><b>then </b>
<font color="Maroon">W</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> 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">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="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</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><b> such that </b><br/><a NAME="E9:109_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">V</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="group_2.html#K4">*</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="txt" href="topgrp_1.html#E3:109_3"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E10:109_3"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E12:109_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="yellow_8.html#T22">YELLOW_8:22</a></i>;<br/>
<a NAME="E13:109_3"/>
( <font color="Maroon">E</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E14:109_3"/><b>then </b>
<font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:109_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="group_2.html#K1">"</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</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#T14" target="_self">Th14</a></i>;<br/>
<b>take </b><font color="Maroon">I</font> = <font color="Maroon">E</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>;<br/>

<a NAME="E18:109_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E20:109_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="group_2.html#K1">"</a>  )
 <b>by </b><i><a class="ref" href="group_2.html#T33">GROUP_2:33</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#D1" target="_self">Def1</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<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"><font color="Maroon">m</font> <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/>.= 
<font color="Maroon">m</font> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E22:109_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E23:109_3"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="topgrp_1.html#T13" target="_self">Th13</a></i>;<br/>

<a NAME="E24:109_3"/>
<font color="Maroon">d</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <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#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E25:109_3"/><b>then </b><i><font color="Green">E150</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">E</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</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_2.html#T34">GROUP_2:34</a></i>;<br/>
<a NAME="E26:109_3"/>
<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> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E27:109_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<b>hence </b><a NAME="E28:109_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>;<br/>

<a NAME="E29:109_3"/>
<font color="Maroon">E</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:109_3_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:109_3_3"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<a NAME="E4:109_3_3"/>
<font color="Maroon">d</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T13" target="_self">Th13</a>, <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/>
<b>hence </b><a NAME="E5:109_3_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="group_2.html#K1">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T13" target="_self">Th13</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/>


</div><b>end;</b></div>
<a NAME="E30:109_3"/><b>then </b>
<font color="Maroon">E</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="group_2.html#K4">*</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:109_3"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font> = 
<font color="Maroon">W</font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>
<br/>.= 
<font color="Maroon">W</font> <a href="group_2.html#K4">*</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#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>
;<br/>
<b>hence </b><a NAME="E33:109_3"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#D2" target="_self">Def2</a>, <a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:109"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">V</font> <a href="group_2.html#K4">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>


</div>
