<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>consider </b><font color="Maroon" title="c4">B</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font>;<br/><b>take </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">B</font>;<br/><b>let </b><font color="Maroon" title="c6">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E3:17_1"/>
<font color="Maroon" title="c6">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ;<br/><a NAME="E4:17_1"/><b>then </b>
( <font color="Maroon" title="c6">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> )
 <b>by </b><i><a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/><b>hence </b><a NAME="E5:17_1"/>
( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">U</font> &amp; <font color="Maroon" title="c6">U</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:17_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="topgen_1.html#T11" target="_self" title="TOPGEN_1:th.11">Th11</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c4">B</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">p</font><b> such that </b><a NAME="E3:17"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/>( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<a NAME="E4:17"/>
for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/>( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:17_2"/>
( <font color="Maroon" title="c5">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:17_2"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c6">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c6">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> )
 <b>by </b><i><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E4:17_2"/><i><font color="Green" title="E10">A4</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:17"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="topgen_1.html#E3:17_2"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E5:17_2"/><b>then </b>
<font color="Maroon" title="c5">U</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="topgen_1.html#E3:17_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:17_2"/>
( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">U</font> &amp; <font color="Maroon" title="c5">U</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="topgen_1.html#E3:17_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="topgen_1.html#E4:17_2"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topgen_1.html#T1" target="_self" title="TOPGEN_1:th.1">Th1</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T61" title="TOPS_1:th.61">TOPS_1:61</a></i>;<br/>


</div>
