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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> non <a href="tdlat_3.html#V3">almost_discrete</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>thus </b><a NAME="E2:91"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a> implies ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a> <a href="tex_3.html#V3" target="_self">boundary</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tex_3.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:91_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:91_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E6:91_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T27">TOPS_3:27</a></i>;<br/>
<a NAME="E7:91_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E4:91_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a> <a href="tex_3.html#V3" target="_self">boundary</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:91_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E5:91_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="tex_3.html#E6:91_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tex_3.html#T67" target="_self">Th67</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E10:91_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E4:91_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="tex_3.html#E9:91_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a> <a href="tex_3.html#V3" target="_self">boundary</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:91"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E1:91_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tex_3.html#E3:91"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tex_3.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a>, <a class="ref" href="tsep_1.html#T11">TSEP_1:11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:91"/><b>then </b>
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T27">TOPS_3:27</a></i>;<br/>
<b>hence </b><a NAME="E7:91"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#D4" target="_self">Def4</a></i>;<br/>


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