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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">T</font> is <a href="urysohn1.html#V1">being_T1</a>
 ;<br/>

<b>assume </b><a NAME="E2:50"/>
<font color="Maroon">A</font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a>
 ;<br/>

<a NAME="E3:50"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font> = 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
<b>by </b><i/>
;<br/>
<a NAME="E5:50"/><i><font color="Green">E42</font></i>: 
 <a href="topgen_1.html#K2" target="_self">Der</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="topgen_1.html#T1" target="_self">Th1</a></i>;<br/>
 <a href="topgen_1.html#K2" target="_self">Der</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> = 
 <a href="topgen_1.html#K2" target="_self">Der</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <span class="p2">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="topgen_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E7:50"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a>
 <b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a></i>;<br/>


</div>
