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<a NAME="E1:72_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T30" title="ZFMISC_1:th.30">ZFMISC_1:30</a></i>;<br/>
<a NAME="E2:72_1_1"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c1">r</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <b>by </b><i><a class="ref" href="relset_1.html#T8" title="RELSET_1:th.8">RELSET_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">C</font> =  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">A</font> =  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">C</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>  <b>by </b><i><a class="txt" href="waybel_9.html#E1:72_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="waybel_9.html#E32"><i><font color="Green" title="E27">Lm4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">A</font> as   <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">A</font>
;<br/>

<b>thus </b><a NAME="E7:72_1_1"/>
( <font color="Maroon" title="c4">A</font> is <a href="waybel_9.html#V1" title="WAYBEL_9:attr.1">strict</a> &amp; not <font color="Maroon" title="c4">A</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> )
 ;<br/>

<b>thus </b><a NAME="E8:72_1_1"/>
<font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">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" title="c5">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c4">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D3" title="COMPTS_1:def.3">COMPTS_1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:72_1_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E2:72_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">F1</font> = <font color="Maroon" title="c5">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ;<br/>
<a NAME="E4:72_1_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c6">F1</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:72_1_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="waybel_9.html#T3" target="_self" title="WAYBEL_9:th.3">Th3</a></i>;<br/>
<a NAME="E5:72_1_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c6">F1</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:72_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="waybel_9.html#T19" target="_self" title="WAYBEL_9:th.19">Th19</a></i>;<br/>
<a NAME="E6:72_1_1_1"/>
 <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T9" title="PCOMPS_1:th.9">PCOMPS_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">G1</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E8:72_1_1_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c7">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F1</font>
 <b>and </b><br/><a NAME="E9:72_1_1_1"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c7">G1</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E10:72_1_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c7">G1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel_9.html#E4:72_1_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="waybel_9.html#E5:72_1_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="compts_1.html#D3" title="COMPTS_1:def.3">COMPTS_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c7">G1</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c4">A</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">G</font>
;<br/>

<b>thus </b><a NAME="E12:72_1_1_1"/>
<font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E8:72_1_1_1"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:72_1_1_1"/>
<font color="Maroon" title="c8">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E9:72_1_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="waybel_9.html#T3" target="_self" title="WAYBEL_9:th.3">Th3</a></i>;<br/>

<b>thus </b><a NAME="E14:72_1_1_1"/>
<font color="Maroon" title="c8">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:72_1_1_1"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a><br/>

<b>assume </b><a NAME="E9:72_1_1"/><i><font color="Green" title="E29">A9</font></i>: 
not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font>
 ;<br/>

<b>assume </b><a NAME="E10:72_1_1"/>
for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">A</font>  holds <br/>(  not <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> or  not <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> or  not <font color="Olive" title="b1">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">V</font> )
 ;<br/>

<a NAME="E11:72_1_1"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:72_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_9.html#E9:72_1_1"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/>


</div>
