<?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">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>, <font color="Maroon">G</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> is <a href="nagata_1.html#V1" target="_self">discrete</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>thus </b><a NAME="E1:7_1"/>
 ex <font color="Olive">O</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> &amp; ( for <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">D</font> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">U</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:7_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font> &amp; <font color="Maroon">U</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">D</font> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">O</font> = <font color="Maroon">U</font>;<br/>

<b>thus </b><a NAME="E3:7_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; ( for <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font> &amp; <font color="Maroon">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">D</font> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:7_1"/>
 ex <font color="Olive">O</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> &amp; ( for <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">D</font> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
<font color="Maroon">F</font> is <a href="nagata_1.html#V1" target="_self">discrete</a>
 <b>by </b><i/>;<br/>


</div>
