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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  <a href="tex_2.html#V2">proper</a> <a href="tex_3.html#V2" target="_self">everywhere_dense</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/>
<a NAME="E2:95"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tex_3.html#T16" target="_self">Th16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E4:95"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V1">dense</a> )
 <b>and </b><br/><a NAME="E5:95"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="tops_1.html#V2">boundary</a> )
 <b>and </b><br/><a NAME="E6:95"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:95"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:95"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tops_3.html#T50">TOPS_3:50</a></i>;<br/>
<a NAME="E9:95"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="tex_3.html#E4:95"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tops_3.html#T17">TOPS_3:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:95"/><i><font color="Green">E34</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>6</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E4:95"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tex_3.html#T23" target="_self">Th23</a></i>;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:95_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="tex_2.html#V1">proper</a>
 ;<br/><a NAME="E2:95_1"/><b>then </b>
<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>1</sub></font>
 <b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/><a NAME="E3:95_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="tex_3.html#E6:95"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:95_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E5:95_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/><b>hence </b><a NAME="E6:95_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <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">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tex_3.html#E11:95"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/>
<a NAME="E14:95"/>
not <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E8:95"><i><font color="Green">E33</font></i></a>, <a class="txt" href="tex_3.html#E12:95"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</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="E16:95"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E5:95"><i><font color="Green">E30</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>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E17:95"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tsep_2.html#R2">constitute_a_decomposition</a> 
 
<b>by </b><i><a class="txt" href="tex_3.html#E7:95"><i><font color="Green">E32</font></i></a>, <a class="txt" href="tex_3.html#E8:95"><i><font color="Green">E33</font></i></a>, <a class="txt" href="tex_3.html#E11:95"><i><font color="Green">E34</font></i></a>, <a class="txt" href="tex_3.html#E16:95"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tsep_2.html#D1">TSEP_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:95"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tsep_2.html#R3">constitute_a_decomposition</a> 
 <b>by </b><i><a class="ref" href="tsep_2.html#D2">TSEP_2:def 2</a></i>;<br/>

<a NAME="E19:95"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tex_3.html#E6:95"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E20:95"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tex_3.html#E11:95"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>


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