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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a> or <font color="Maroon">c<sub>3</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a> )
 ;<br/>

<b>assume </b><a NAME="E2:57"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a> or <font color="Maroon">c<sub>3</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a> )
 </span><b>by </b><i><a class="txt" href="tex_3.html#E1:57"><i><font color="Green">E27</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 ;<br/><div class="add"><a NAME="E2:57_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:57"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E1:57_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tex_3.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 ;<br/><div class="add"><a NAME="E2:57_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:57"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tsep_1.html#T30">TSEP_1:30</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>3</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E1:57_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tex_3.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


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