<?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/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<a NAME="E1:81"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">A</font> is <a href="isomichi.html#V3" target="_self">1st_class</a> implies <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="isomichi.html#V3" target="_self">1st_class</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon">A</font> is <a href="isomichi.html#V3" target="_self">1st_class</a>
 ;<br/>

<a NAME="E2:81_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">A</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">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isomichi.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:81_1"/><b>then </b>
<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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<a NAME="E4:81_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4">TDLAT_3:4</a></i>;<br/>
<a NAME="E5:81_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T3">TDLAT_3:3</a></i>;<br/>
<a NAME="E6:81_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4">TDLAT_3:4</a></i>;<br/>
<a NAME="E7:81_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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T3">TDLAT_3:3</a></i>;<br/>
<b>hence </b><a NAME="E8:81_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="isomichi.html#V3" target="_self">1st_class</a>
 <b>by </b><i><a class="ref" href="isomichi.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:81"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="isomichi.html#V3" target="_self">1st_class</a> implies <font color="Maroon">A</font> is <a href="isomichi.html#V3" target="_self">1st_class</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="isomichi.html#V3" target="_self">1st_class</a>
 ;<br/>

<a NAME="E2:81_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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="isomichi.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:81_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T3">TDLAT_3:3</a></i>;<br/>
<a NAME="E4:81_2"/><b>then </b>
<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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4">TDLAT_3:4</a></i>;<br/>
<a NAME="E5:81_2"/><b>then </b>
<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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4">TDLAT_3:4</a></i>;<br/>
<a NAME="E6:81_2"/><b>then </b>
<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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <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">A</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T3">TDLAT_3:3</a></i>;<br/>
<a NAME="E7:81_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">A</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">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<b>hence </b><a NAME="E8:81_2"/>
<font color="Maroon">A</font> is <a href="isomichi.html#V3" target="_self">1st_class</a>
 <b>by </b><i><a class="ref" href="isomichi.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/>
( <font color="Maroon">A</font> is <a href="isomichi.html#V3" target="_self">1st_class</a> iff <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="isomichi.html#V3" target="_self">1st_class</a> )
 <b>by </b><i/>;<br/>


</div>
