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<div class="add">

<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>assume </b><a NAME="E1:41"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">T</font> is <a href="group_1.html#V6">finite</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<a NAME="E2:41"/>
<span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> : <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">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">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:41_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> : <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:41_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>
<b>thus </b><a NAME="E5:41_1"/>
<font color="Maroon">z</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">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">sfA</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> : <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">sfA</font> = <font color="Maroon">sfA</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sfA</font>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:41_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>thus </b><a NAME="E5:41_2"/>
<font color="Maroon">P</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/></div><b>end;</b></div>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">sfA</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="E7:41"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">tT</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:41"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">tT</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="pre_circ.html#S1">PRE_CIRC:sch 1</a>();<br/></i>
<a NAME="E10:41"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">sfA</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/>
<a NAME="E11:41"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is    <a href="subset_1.html#M2">Element</a> of 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#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E12:41"/><b>then </b><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sfA</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sfA</font>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:41_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>thus </b><a NAME="E5:41_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E14:41"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">sfA</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E16:41"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E17:41"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E18:41"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">sfA</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<a NAME="E19:41"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">sfA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:41_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">sfA</font>
 ;<br/>

<a NAME="E2:41_4"/>
<font color="Maroon">B</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><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E3:41_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sfA</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:41_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:41"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> : <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
