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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">D</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> holds <br/> ex <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Olive" title="b3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b2">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b2">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">B</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="c1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">D</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> implies  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</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="c1">X</font> ) )</span><br/>



<b>assume </b><a NAME="E1:48"/>
<font color="Maroon" title="c2">D</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</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="c1">X</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E4:48"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E5:48"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T27" target="_self" title="TOPS_3:th.27">Th27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Olive" title="b1">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">B</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="c1">X</font> )</span><br/>

<b>take </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</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="c1">X</font> )</span><br/>

<b>thus </b><a NAME="E6:48"/>
( <font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>by </b><i><a class="txt" href="tops_3.html#E4:48"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="tops_3.html#E5:48"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a>, <a class="ref" href="tops_1.html#D4" title="TOPS_1:def.4">TOPS_1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</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="c1">X</font> )</span><br/>

<a NAME="E7:48"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span>
<b>by </b><i><a class="txt" href="tops_3.html#E7:48"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">D</font>
<b>by </b><i><a class="txt" href="tops_3.html#E3:48"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</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="c1">X</font> )</span><br/>

<a NAME="E9:48"/>
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E10:48"/>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</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="c1">X</font></span><br/>

<a NAME="E11:48"/>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/>
<b>hence </b><a NAME="E12:48"/>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</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="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
