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<b>let </b><font color="Maroon">G</font> be   <a href="topgrp_1.html#NM5">TopologicalGroup</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>let </b><font color="Maroon">A</font> be    <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"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>;<br/>





<b>consider </b><font color="Maroon">X</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">Y</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><b> such that </b><br/><a NAME="E2:103"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">Y</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="txt" href="topgrp_1.html#E6:103"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <b>by </b><i><a class="ref" href="connsp_2.html#T4">CONNSP_2:4</a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</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="E4:103"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font>, <font color="Maroon">h</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:103"/><i><font color="Green">E40</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">B</font> &amp; <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><a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<a NAME="E7:103"/>
<font color="Maroon">h</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:103"/><b>then </b>
<font color="Maroon">h</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:103"/><b>then </b>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="group_2.html#K1">"</a>  )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:103"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">Y</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="E11:103"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


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