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<b>let </b><font color="Maroon">G</font> be  <a href="topgrp_1.html#V4" target="_self">BinContinuous</a> <a href="topgrp_1.html#NM4">TopGroup</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Maroon">O</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:99"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">O</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:99"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><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="E2:99_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:99_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <font color="Maroon">O</font> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>;<br/>
<a NAME="E5:99_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">O</font> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T9" target="_self">Th9</a>, <a class="txt" href="topgrp_1.html#E5:99_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E6:99_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">O</font> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><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="E1:99_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:99_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<b>thus </b><a NAME="E4:99_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:99_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <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="E8:99_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:99"/>
<font color="Maroon">O</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


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