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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/>

<b>consider </b><font color="Maroon">B</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/>
<b>consider </b><font color="Maroon">B1</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="topgen_2.html#T20">TOPGEN_2:20</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a<sub>1</sub></font> =  <a href="subset_1.html#K10">choose</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">BB</font>) where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : ( <font color="Olive">BB</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">BB</font>] ) </span>}</span> ;<br/>
<a NAME="E4:12"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">BB</font>) where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : ( <font color="Olive">BB</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">BB</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">BB</font>) where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : ( <font color="Olive">BB</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">BB</font>] ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">B2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>
<a NAME="E4:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">BB</font>) where <font color="Olive">BB</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : ( <font color="Olive">BB</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">BB</font>] ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E6:12"/>
for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:12_2"/>
( <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E4:12_2"/><i><font color="Green">E52</font></i>: 
 <a href="subset_1.html#K10">choose</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:12_2"/>
 <a href="subset_1.html#K10">choose</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:12_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:12"/><b>then </b>
<font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/>
<a NAME="E8:12"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="topgen_1.html#K4">density</a> <font color="Maroon">T</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D12">TOPGEN_1:def 12</a></i>;<br/>
<a NAME="E9:12"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : ( <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] ) </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B1</font>
 
<b>from </b><i><a class="ref" href="borsuk_2.html#S1">BORSUK_2:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E10:12"/>
 <a href="topgen_1.html#K4">density</a> <font color="Maroon">T</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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