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<b>let </b><font color="Maroon">T</font> be   <a href="topgrp_1.html#NM5">TopologicalGroup</a>;<br/><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> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>;<br/>





<b>consider </b><font color="Maroon">A</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">B</font> being  <a href="pre_topc.html#V3">open</a>  <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="E2:80"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>
<b>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 <font color="Maroon">b</font><b> such that </b><br/><a NAME="E4:80"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">C</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">C</font>
;<br/>

<a NAME="E5:80"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2">GROUP_2:def 2</a></i>;<br/>
<a NAME="E6:80"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="group_2.html#K1">"</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2">GROUP_2:def 2</a></i>;<br/>
<a NAME="E7:80"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">C</font> <a href="group_2.html#K1">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K3">"</a> </span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1">GROUP_2:def 1</a></i>;<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="E8:80"/>
<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">C</font> <a href="group_2.html#K1">"</a> </span>)</span>
 ;<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="E10:80"/><i><font color="Green">E52</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">C</font> <a href="group_2.html#K1">"</a>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:80"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="topgrp_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E13:80"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</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> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="topgrp_1.html#T7" target="_self">Th7</a>, <a class="txt" href="topgrp_1.html#E5:80"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:80"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a>, <a class="ref" href="topgrp_1.html#T7" target="_self">Th7</a>, <a class="txt" href="topgrp_1.html#E5:80"><i><font color="Green">E47</font></i></a></i>;<br/>


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