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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] );<br/><b>consider </b><font color="Maroon" title="c1">G</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E4:26_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>hence </b><a NAME="E2:26_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c2">GA</font> = <font color="Maroon" title="c1">G</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><a NAME="E6:26_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E7:26_1_1"/><b>then </b>
<font color="Maroon" title="c2">GA</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="finset_1.html#E2:26"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="finset_1.html#E3:26_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:26_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">GA</font> &amp; ( for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">GA</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E1:26"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="finset_1.html#T18" target="_self" title="FINSET_1:th.18">Th18</a></i>;<br/><a NAME="E10:26_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  ex <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="finset_1.html#E9:26_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">B</font>;<br/><b>assume </b><a NAME="E2:26_1_1_2"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E3:26_1_1_2"/><b>then </b>
not <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E9:26_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:26_1_1_2"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E5:26_1_1_2"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:26_1_1_2"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="finset_1.html#E10:26_1_1"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/><a NAME="E7:26_1_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="finset_1.html#E5:26_1_1_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E8:26_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="finset_1.html#E9:26_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E9:26_1_1_2"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">GA</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="finset_1.html#E6:26_1_1_2"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/><a NAME="E10:26_1_1_2"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E4:26_1_1_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:26_1_1_2"/><b>then </b>
not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E12:26_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E13:26_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="finset_1.html#E9:26_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="finset_1.html#E9:26_1_1_2"><i><font color="Green" title="E18">A10</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:26_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>()]
 <b>by </b><i><a class="txt" href="finset_1.html#E10:26_1_1"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>()]
 <b>by </b><i><a class="txt" href="finset_1.html#E2:26"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>


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