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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">T</font> <a href="compts_1.html#NR3" target="_self">is_T3</a> 
 ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E2:44_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E3:44_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="yellow_8.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> or <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">Q</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>;<br/><b>thus </b><a NAME="E2:44_1_1_1"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><b>thus </b><a NAME="E3:44_1_1_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E4:44_1_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T43">TOPS_1:43</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Q</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><div class="add"><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:44_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E3:44_1_1_2_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>thus </b><a NAME="E4:44_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:44_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:44_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:44_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E7:44_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E8:44_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i>, , , <a class="ref" href="compts_1.html#D5">COMPTS_1:def 5</a></i>;<br/><b>take </b><font color="Maroon">Q</font> = <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> ;<br/><b>thus </b><a NAME="E9:44_1_1_2"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/><a NAME="E10:44_1_1_2"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><a NAME="E11:44_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/><a NAME="E12:44_1_1_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E13:44_1_1_2"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E1:44_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:44_1_1_2"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E15:44_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/><b>hence </b><a NAME="E16:44_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:44"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><br/> for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">P</font> holds <br/> ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">Q</font> )
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D5">COMPTS_1:def 5</a><br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:44"/>
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E5:44"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E6:44"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#T4" target="_self">Th4</a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E7:44"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:44"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E10:44"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E11:44"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>take </b><font color="Maroon">V</font> = <font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> ;<br/>

<b>thus </b><a NAME="E13:44"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>thus </b><a NAME="E14:44"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>

<b>thus </b><a NAME="E15:44"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>

<a NAME="E16:44"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<b>hence </b><a NAME="E17:44"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E18:44"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">Q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E19:44"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E20:44"/>
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


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