<?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="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:105_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E2:105_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V4" target="_self">condensed</a>
 ;<br/><a NAME="E3:105_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tops_1.html#E1:105_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E4:105_1"/><i><font color="Green">E68</font></i>: 
(  <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <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" target="_self">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="tops_1.html#E2:105_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E5:105_1"/>
 <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tops_1.html#E2:105_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tops_1.html#E3:105_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E6:105_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tops_1.html#E3:105_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E7:105_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tops_1.html#E4:105_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E8:105_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V5" target="_self">closed_condensed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D7" target="_self">Def7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:105"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V5" target="_self">closed_condensed</a>
 ;<br/>

<a NAME="E3:105"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:105"/>
 <a href="tops_1.html#K2" target="_self">Fr</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tops_1.html#E80"><i><font color="Green">Lemma39</font></i></a></i>;<br/>
<b>then </b> <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>3</sub></font> = 
<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" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="tops_1.html#E2:105"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tops_1.html#T102" target="_self">Th102</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tops_1.html#E3:105"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tops_1.html#T45" target="_self">Th45</a></i>
;<br/>
<a NAME="E6:105"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T77" target="_self">Th77</a></i>;<br/>
<a NAME="E7:105"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E8:105"/><b>then </b>
 <a href="tops_1.html#K1" target="_self">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E9:105"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V4" target="_self">condensed</a> )
 <b>by </b><i><a class="txt" href="tops_1.html#E3:105"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self">Def6</a></i>;<br/>


</div>
