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

<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P</font> is <a href="tops_1.html#V1" target="_self">dense</a>
 ;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:70_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:70_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E4:70_1"/><i><font color="Green">E37</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font>;<br/><a NAME="E6:70_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E7:70_1"/><b>then </b><i><font color="Green">E38</font></i>: 
not <font color="Maroon">TS</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E8:70_1"/>
<font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">TS</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:70_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T45" target="_self">Th45</a>, <a class="ref" href="pre_topc.html#T13">PRE_TOPC:13</a></i>;<br/><b>hence </b><a NAME="E10:70_1"/>
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tops_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tops_1.html#T45" target="_self">Th45</a>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:70"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>  st <font color="Olive">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Q</font>
 ;<br/>

<a NAME="E3:70"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<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="E1:70_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font>
 ;<br/>

<a NAME="E2:70_2"/><b>then </b><i><font color="Green">E41</font></i>: 
not <font color="Maroon">TS</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E3:70_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>  st <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> holds <br/><font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:70_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tops_1.html#T44" target="_self">Th44</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:70"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">TS</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:70"/>
<font color="Maroon">P</font> is <a href="tops_1.html#V1" target="_self">dense</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T40" target="_self">Th40</a></i>;<br/>


</div>
