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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">F</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">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> ;<br/>
<a NAME="E1:10"/><i><font color="Green">E13</font></i>: 
<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">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<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">C</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="E1:10_1"/>
<font color="Maroon">C</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">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:10_1"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:10"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</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">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">F</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">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:10_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E4:10_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:10_2"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><b>thus </b><a NAME="E6:10_2"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E7:10"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_3"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:10_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:10_3"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:10_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><b>thus </b><a NAME="E6:10_3"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E9:10"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E10:10"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<a NAME="E11:10"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E12:10"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E13:10"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E14:10"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</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">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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