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

<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<a NAME="E1:69"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="topgen_1.html#V6" target="_self">scattered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1"/>
not  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="topgen_1.html#V6" target="_self">scattered</a>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> &amp; <font color="Maroon">B</font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a> )
 <b>by </b><i><a class="txt" href="topgen_1.html#E3:69_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:69_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:69"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> is <a href="topgen_1.html#V6" target="_self">scattered</a>
 ;<br/>


</div>
