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<b>let </b><font color="Maroon" title="c1">T</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>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">W</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="c3">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>;<br/>





<b>consider </b><font color="Maroon" title="c5">A</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>, <font color="Maroon" title="c6">B</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="c3">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> <b> such that </b><br/><a NAME="E2:80"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T37" target="_self" title="TOPGRP_1:th.37">Th37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">C</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="c3">b</font><b> such that </b><br/><a NAME="E4:80"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T39" target="_self" title="TOPGRP_1:th.39">Th39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">C</font>
;<br/>

<a NAME="E5:80"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2" title="GROUP_2:def.2">GROUP_2:def 2</a></i>;<br/>
<a NAME="E6:80"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font></span>)</span> where <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2" title="GROUP_2:def.2">GROUP_2:def 2</a></i>;<br/>
<a NAME="E7:80"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1" title="GROUP_2:def.1">GROUP_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">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="E8:80"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E10:80"/><i><font color="Green" title="E46">A6</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E6:80"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E12:80"/><i><font color="Green" title="E47">A7</font></i>: 
( <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E7:80"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="topgrp_1.html#E10:80"><i><font color="Green" title="E46">A6</font></i></a></i>;<br/>
<a NAME="E13:80"/>
<font color="Maroon" title="c9">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E4:80"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="topgrp_1.html#E5:80"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="topgrp_1.html#E10:80"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="topgrp_1.html#E12:80"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:80"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:80"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="topgrp_1.html#E10:80"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="topgrp_1.html#E12:80"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/>


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