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<b>consider </b><font color="Maroon">K</font> being    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:27_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>]
 <b>by </b><i/>;<br/>
<a NAME="E3:27_1"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">K</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<a NAME="E4:27_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E5:27_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">F<sub>1</sub></font>()
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>

<b>then consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E7:27_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">K</font> &amp; <font color="Maroon">A</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="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:27_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E4:27_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E5:27_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">a</font>];<br/><a NAME="E6:27_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><a NAME="E8:27_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E9:27_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 <b>by </b><i/>;<br/><a NAME="E10:27_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Z</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">xx</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">Z'</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">xx</font>, <font color="Maroon">Z'</font> = <font color="Maroon">Z'</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">xx</font>, <font color="Maroon">Z'</font> = <font color="Maroon">Z'</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">Ax</font> = <font color="Maroon">x</font>, <font color="Maroon">Ay</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Z'</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="waybel19.html#E1:27"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E6:27_1_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">Ax</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">Ay</font>] )
 
<b>by </b><i>, , <a class="txt" href="waybel19.html#E1:27"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E7:27_1_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ax</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T9">MSSUBFAM:9</a></i>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E8:27_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E9:27_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ax</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Ay</font>
 
<b>by </b><i>, <a class="ref" href="mssubfam.html#T8">MSSUBFAM:8</a></i>;<br/>
<b>hence </b><a NAME="E10:27_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">a</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E11:27_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Y</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="ref" href="waybel19.html#D1" target="_self">Def1</a>, <a class="txt" href="waybel19.html#E2"><i><font color="Green">Lemma28</font></i></a>, );<br/></i><b>hence </b><a NAME="E12:27_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:27_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i>, </i>;<br/>


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