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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> non <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</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="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">D</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:93"/><i><font color="Green" title="E29">A1</font></i>: 
(  not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> or  not <font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> )
 ;<br/>

<a NAME="E3:93"/>
<font color="Maroon" title="c3">D</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tex_3.html#T16" target="_self" title="TEX_3:th.16">Th16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">B</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="E5:93"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c4">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>and </b><br/><a NAME="E6:93"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>and </b><br/><a NAME="E7:93"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 <b>and </b><br/><a NAME="E8:93"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="tops_3.html#T49" title="TOPS_3:th.49">TOPS_3:49</a></i>;<br/>
<a NAME="E9:93"/>
<font color="Maroon" title="c4">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> 
 
<b>by </b><i><a class="txt" href="tex_3.html#E5:93"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="tops_3.html#T17" title="TOPS_3:th.17">TOPS_3:17</a></i>;<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="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</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="E11:93"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c4">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="txt" href="tex_3.html#E5:93"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="tex_3.html#T23" target="_self" title="TEX_3:th.23">Th23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1"/>
not <font color="Maroon" title="c4">C</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ;<br/><a NAME="E2:93_1"/><b>then </b><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c4">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="c1">X</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T5" title="TEX_2:th.5">TEX_2:5</a></i>;<br/><a NAME="E3:93_1"/>
( <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c3">D</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="c1">X</font> )
 <b>by </b><i><a class="txt" href="tex_3.html#E7:93"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:93_1"/><b>then </b>
<font color="Maroon" title="c3">D</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="txt" href="tex_3.html#E2:93_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E5:93_1"/><b>then </b>
( <font color="Maroon" title="c3">D</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c3">D</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a>, <a class="ref" href="tops_3.html#T16" title="TOPS_3:th.16">TOPS_3:16</a></i>;<br/><b>hence </b><a NAME="E6:93_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E2:93"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">X1</font> = <font color="Maroon" title="c6">X1</font> as  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="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a> <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tex_3.html#E11:93"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_2_1"/>
(  not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> or  not <font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> )
 </span><b>by </b><i><a class="txt" href="tex_3.html#E2:93"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_2_1_1"><b>suppose </b></a><a NAME="E1:93_2_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
not <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:93_2_1_1"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>thus </b><a NAME="E3:93_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E1:93_2_1_1"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_2_1_2"><b>suppose </b></a><a NAME="E1:93_2_1_2"/><i><font color="Green" title="E35">A9</font></i>: 
not <font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:93_2_1_2"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:93_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E5:93"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="tex_3.html#E7:93"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="tex_3.html#E1:93_2_1_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">X2</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="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</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="E16:93"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c5">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="c8">X2</font>
 <b>by </b><i><a class="txt" href="tex_3.html#E6:93"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="tex_3.html#T62" target="_self" title="TEX_3:th.62">Th62</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">X1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">X2</font>
;<br/>

<b>thus </b><a NAME="E17:93"/>
<font color="Maroon" title="c7">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c8">X2</font>
 <b>by </b><i><a class="txt" href="tex_3.html#E8:93"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="tex_3.html#E11:93"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="tex_3.html#E16:93"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></i>;<br/>

<a NAME="E18:93"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c8">X2</font></span>)</span> <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="c2">X0</font>
 
<b>by </b><i><a class="txt" href="tex_3.html#E7:93"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="tex_3.html#E11:93"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="tex_3.html#E16:93"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="tsep_1.html#D2" title="TSEP_1:def.2">TSEP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E19:93"/>
<font color="Maroon" title="c7">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c8">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">X0</font> #)
 <b>by </b><i><a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/>


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