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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> is <a href="finset_1.html#V1">finite</a> );<br/>
<b>consider </b><font color="Maroon">P</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:17_1_1"/><i><font color="Green">E13</font></i>: 
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">P</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:17_1_1"/>
 <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="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:17_1_1"/>
for <font color="Olive">X</font>, <font color="Olive">Y</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">Q</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/>( <font color="Olive">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">X</font> <a href="subset_1.html#K6">\</a> <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_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">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Z1</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">Z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:17_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">Z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Z2</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a>, <a class="ref" href="finsub_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:17_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z2</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="finset_1.html#T14">FINSET_1:14</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E7:17_1_1_1"/>
( <font color="Maroon">Z1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">Z1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z2</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#T10" target="_self">Th10</a>, <a class="ref" href="finset_1.html#T16">FINSET_1:16</a></i>;<br/>
<a NAME="E8:17_1_1_1"/><b>then </b>
( <font color="Maroon">Z1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Z1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Z1</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z2</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:17_1_1_1"/>
( <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a>, <a class="txt" href="finsub_1.html#E2:17_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">Q</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finsub_1.html#V4" target="_self">preBoolean</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<a NAME="E7:17_1_1"/>
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">R</font> iff ( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">X</font> is <a href="finset_1.html#V1">finite</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:17_1_1_2"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> implies ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:17_1_1_2_1"/><b>then </b>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">Y</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1_1_2_1"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:17_1_1_2"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> implies <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:17_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="finsub_1.html#V4" target="_self">preBoolean</a>  <a href="hidden.html#M1">set</a>  st <br/>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="Olive">b<sub>1</sub></font> iff ( <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">X</font> is <a href="finset_1.html#V1">finite</a> ) )
 ;<br/>


</div>
