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<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="E1:106_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:106_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="E3:106_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tops_1.html#E1:106_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tops_1.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E4:106_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#E2:106_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="E5:106_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E6:106_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <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>
 <b>by </b><i><a class="txt" href="tops_1.html#E3:106_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tops_1.html#T48" target="_self">Th48</a></i>;<br/><a NAME="E7:106_1"/><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>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#E4:106_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="E8:106_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V6" target="_self">open_condensed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D8" target="_self">Def8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:106"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V6" target="_self">open_condensed</a>
 ;<br/>

<a NAME="E3:106"/><b>then </b>
(  <a href="tops_1.html#K2" target="_self">Fr</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K2" target="_self">Fr</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> &amp;  <a href="tops_1.html#K2" target="_self">Fr</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="hidden.html#R1">=</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="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T104" target="_self">Th104</a></i>;<br/>
<a NAME="E4:106"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E5:106"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i><a class="txt" href="tops_1.html#E2:106"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tops_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E6:106"/>
 <a href="tops_1.html#K1" target="_self">Int</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="txt" href="tops_1.html#E4:106"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tops_1.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E7:106"/><b>then </b>
<font color="Maroon">c<sub>3</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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E8:106"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</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#E5:106"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tops_1.html#D6" target="_self">Def6</a></i>;<br/>


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