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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> ;<br/>
<a NAME="E2:10"/><i><font color="Green">E20</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E3:10"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> iff  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E2:10_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:10_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:10_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:10_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E8:10_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>take </b><font color="Maroon">W</font> = <font color="Maroon">W</font>;<br/><b>thus </b><a NAME="E9:10_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_8.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E11:10_1_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="yellow_8.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E12:10_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="yellow_8.html#T1" target="_self">Th1</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:10_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:10_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 ;<br/>

<a NAME="E4:10_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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