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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#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  ex <font color="Olive">B</font> being   <a href="yellow_8.html#M1" target="_self">Basis</a> of <font color="Olive">x</font> st <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/>

<b>thus </b><a NAME="E3:15"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a><br/>

<b>let </b><font color="Maroon">e</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="E4:15"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> ) </span>}</span> ;<br/>
<a NAME="E6:15"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:15_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
;<br/>
<b>hence </b><a NAME="E3:15_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:15"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff  ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:15_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>consider </b><font color="Maroon">B</font> being    <a href="yellow_8.html#M1" target="_self">Basis</a> of <font color="Maroon">p</font><b> such that </b><br/><a NAME="E5:15_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><b>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="E7:15_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:15_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">W</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font>;<br/><b>thus </b><a NAME="E10:15_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E1:15"><i><font color="Green">E19</font></i></a>, </i>;<br/><b>thus </b><a NAME="E11:15_2_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="yellow_8.html#E1:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow_8.html#E2:15"><i><font color="Green">E20</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:15_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E3:15_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:15_2"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:15_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:15"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:15"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T1" target="_self">Th1</a>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/>


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