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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>;<br/>







<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:72"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ;<br/>

<b>assume </b><a NAME="E3:72"/><i><font color="Green" title="E45">A3</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> )
 ;<br/>

<b>assume </b><a NAME="E4:72"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 ;<br/>

<div><i><font color="Green" title="E47">A5</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" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:72_1"/><b>then </b>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X0</font>
 <b>by </b><i><a class="txt" href="tops_3.html#E3:72"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">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" title="c3">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">X1</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:72_2"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">X1</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10" title="TSEP_1:th.10">TSEP_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">E</font> = <font color="Maroon" title="c5">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">X1</font> <b>by </b><i><a class="txt" href="tops_3.html#E3:72"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="tops_3.html#E3:72_2"><i><font color="Green" title="E48">A6</font></i></a></i>;<br/><a NAME="E5:72_2"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c7">E</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E1:72"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="tops_3.html#E3:72"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="tops_3.html#E4:72"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="tops_3.html#E3:72_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="tops_3.html#E74"><i><font color="Green" title="E41">Lm1</font></i></a></i>;<br/><a NAME="E6:72_2"/>
<font color="Maroon" title="c6">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</font>
 <b>by </b><i><a class="txt" href="tops_3.html#E2:72"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="tops_3.html#E3:72_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><b>hence </b><a NAME="E7:72_2"/>
<font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E5:72_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="tops_3.html#T68" target="_self" title="TOPS_3:th.68">Th68</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:72"/>
<font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E5:72"><i><font color="Green" title="E47">A5</font></i></a></i>;<br/>


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