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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>thus </b><a NAME="E1:57"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V3">almost_discrete</a> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V3">almost_discrete</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="tex_1.html#T37" target="_self">Th37</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> ;<br/>

<b>thus </b><a NAME="E4:57_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="tex_1.html#E3:57_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a>, <a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a>, <a class="ref" href="tops_3.html#T1">TOPS_3:1</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:57"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tex_1.html#E2:57"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tops_3.html#T2">TOPS_3:2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><a NAME="E2:57_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tex_1.html#E3:57"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a>, <a class="ref" href="tops_3.html#T18">TOPS_3:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V3">almost_discrete</a>
 <b>by </b><i><a class="ref" href="tex_1.html#T37" target="_self">Th37</a></i>;<br/>


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