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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font>;<br/>







<b>assume </b><a NAME="E1:72"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:72"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>

<b>assume </b><a NAME="E3:72"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/>

<b>assume </b><a NAME="E4:72"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:72_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon">B</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_2"/>
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">X1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:72_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10">TSEP_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font> <b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self">Th39</a>, <a class="txt" href="tops_3.html#E5:72_2"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E5:72_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">E</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T36" target="_self">Th36</a>, <a class="ref" href="tops_3.html#T39" target="_self">Th39</a>, <a class="ref" href="tops_3.html#T40" target="_self">Th40</a>, <a class="txt" href="tops_3.html#E5:72_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tops_3.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:72_2"/>
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="tops_3.html#T38" target="_self">Th38</a>, <a class="txt" href="tops_3.html#E5:72_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>hence </b><a NAME="E7:72_2"/>
<font color="Maroon">B</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i>, <a class="ref" href="tops_3.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:72"/>
<font color="Maroon">B</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T41" target="_self">Th41</a></i>;<br/>


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