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<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">K</font>
 ;<br/><b>take </b><font color="Maroon">Q</font> =  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">K</font>;<br/><b>thus </b><a NAME="E2:39_1"/>
( <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T29" target="_self">Th29</a>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><a NAME="E2:39"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E5:39"/>
<font color="Maroon">K</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#T30" target="_self">Th30</a>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<a NAME="E6:39"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <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"><font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E7:39"/>
<font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29" target="_self">Th29</a>, </i>;<br/>
<a NAME="E8:39"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tops_1.html#T34" target="_self">Th34</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E9:39"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="subset_1.html#K3">`</a> </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="E10:39"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">K</font>
 
;<br/>
<b>hence </b><a NAME="E11:39"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1" target="_self">Int</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T32" target="_self">Th32</a></i>;<br/>


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