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<a NAME="E1:72_1_1"/><i><font color="Green">E33</font></i>: 
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp;  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T30">ZFMISC_1:30</a></i>;<br/>
<a NAME="E2:72_1_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="relset_1.html#T8">RELSET_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="waybel_9.html#G1" target="_self">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">r</font>,<font color="Maroon">C</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="realset2.html#V3">trivial</a> <a href="tdlat_3.html#V1">discrete</a> <a href="orders_2.html#V2">reflexive</a>  <a href="waybel_9.html#L1" target="_self">TopRelStr</a>  <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="waybel_9.html#NM1">TopLattice</a> ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E7:72_1_1"/>
( <font color="Maroon">A</font> is <a href="waybel_9.html#V1" target="_self">strict</a> &amp; not <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">A</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">A</font> is <a href="tdlat_3.html#V1">discrete</a> &amp; <font color="Maroon">A</font> is <a href="group_1.html#V6">finite</a> )
 ;<br/>

<b>thus </b><a NAME="E8:72_1_1"/>
<font color="Maroon">A</font> is <a href="compts_1.html#V2">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:72_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:72_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span> ;<br/>
<a NAME="E4:72_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F1</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:72_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">F1</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E6:72_1_1_1"/>
 <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T9">PCOMPS_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">G1</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E8:72_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1</font>
 <b>and </b><br/><a NAME="E9:72_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">G1</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E10:72_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">G1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G1</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E12:72_1_1_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E1:72_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:72_1_1_1"/>
<font color="Maroon">G</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E1:72_1_1_1"><i><font color="Green">E34</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E14:72_1_1_1"/>
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="waybel_9.html#E2:72_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">A</font>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a><br/>

<b>assume </b><a NAME="E9:72_1_1"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<b>assume </b><a NAME="E10:72_1_1"/>
for <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font>  holds <br/>(  not <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> or  not <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> or  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> or  not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> or  not <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">V</font> )
 ;<br/>

<a NAME="E11:72_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:72_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


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