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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:72"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="isomichi.html#V3" target="_self">1st_class</a> implies  <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="isomichi.html#V3" target="_self">1st_class</a>
 ;<br/>

<a NAME="E2:72_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="isomichi.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:72_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:72_1"/>
 <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="isomichi.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:72"/>
(  <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="isomichi.html#V3" target="_self">1st_class</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_2"/>
 <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E2:72_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="isomichi.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E3:72_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:72_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="isomichi.html#V3" target="_self">1st_class</a>
 <b>by </b><i><a class="ref" href="isomichi.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:72"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="isomichi.html#V3" target="_self">1st_class</a> iff  <a href="isomichi.html#K1" target="_self">Border</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="txt" href="isomichi.html#E1:72"><i><font color="Green">E39</font></i></a></i>;<br/>


</div>
