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<b>let </b><font color="Maroon">X</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">X0</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tex_3.html#V2" target="_self">everywhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">D</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:93"/><i><font color="Green">E16</font></i>: 
(  not <font color="Maroon">X0</font> is <a href="tex_3.html#V1" target="_self">dense</a> or  not <font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a> )
 ;<br/>

<a NAME="E3:93"/>
<font color="Maroon">D</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:93"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> is <a href="tops_1.html#V1">dense</a> )
 <b>and </b><br/><a NAME="E6:93"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>and </b><br/><a NAME="E7:93"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E8:93"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tops_3.html#T49">TOPS_3:49</a></i>;<br/>
<a NAME="E9:93"/>
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="tops_3.html#T17">TOPS_3:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">X1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1">open</a> <a href="tex_3.html#V1" target="_self">dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:93"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i>, </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">C</font> is <a href="tex_2.html#V1">proper</a>
 ;<br/><a NAME="E2:93_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/><a NAME="E3:93_1"/>
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tex_3.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:93_1"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="tex_3.html#E7:93"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E5:93_1"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E6:93_1"/><b>then </b>
( <font color="Maroon">D</font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a>, <a class="ref" href="tops_3.html#T16">TOPS_3:16</a></i>;<br/><b>hence </b><a NAME="E7:93_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T16">TSEP_1:16</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X1</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1">open</a> <a href="tex_2.html#V2">proper</a> <a href="tex_3.html#V1" target="_self">dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="tex_3.html#E6:93"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tex_2.html#T13">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">X0</font> is <a href="tex_3.html#V1" target="_self">dense</a> or  not <font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a> )
 </span><b>by </b><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">E50</font></i>: 
not <font color="Maroon">X0</font> is <a href="tex_3.html#V1" target="_self">dense</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:93_2_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E3:93_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E7:93"><i><font color="Green">E35</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">E61</font></i>: 
not <font color="Maroon">X0</font> is <a href="tsep_1.html#V1">open</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:93_2_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:93_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tex_3.html#T9" target="_self">Th9</a>, <a class="txt" href="tex_3.html#E7:93"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tsep_1.html#T16">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">X2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tex_3.html#V4" target="_self">nowhere_dense</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E16:93"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tex_3.html#D1" target="_self">Def1</a>, <a class="ref" href="tex_3.html#T8" target="_self">Th8</a></i>;<br/>
<b>take </b>
<font color="Maroon">X1</font>
;<br/>

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

<b>thus </b><a NAME="E17:93"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="txt" href="tex_3.html#E6:93"><i><font color="Green">E34</font></i></a>, <a class="txt" href="tex_3.html#E7:93"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>

<a NAME="E18:93"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 
<b>by </b><i><a class="ref" href="tex_3.html#T9" target="_self">Th9</a>, <a class="txt" href="tex_3.html#E6:93"><i><font color="Green">E34</font></i></a>, <a class="txt" href="tex_3.html#E7:93"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E19:93"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #)
 <b>by </b><i><a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/>


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