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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">R</font> be    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>

<i><font color="Green">E34</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
<b>by </b><i/>
;<br/>
<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E3:32"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 ;<br/>

<b>assume </b><a NAME="E4:32"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E5:32"/><b>then </b>
<font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
<font color="Maroon">W</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon">W</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:32"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


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