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<a NAME="E1:74_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F<sub>4</sub></font>() is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E2:74_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:74"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E3:74_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">B</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">F<sub>4</sub></font>() &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</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="74_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">C1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">C1</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">C1'</font> = <font color="Maroon">C1</font> as   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> or  not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">C1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_2"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:74_1_1_1_2_1"/>
<font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:74_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:74_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:74_1_1_1_2"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>then </b><i><font color="Green">E64</font></i>: <font color="Maroon">C1'</font> <a href="chain_1.html#K8" target="_self">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A'</font></span><a href="domain_1.html#K6">}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T101">XBOOLE_1:101</a></i>
<br/>.= 
<font color="Maroon">C1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>

;<br/><a NAME="E5:74_1_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">A'</font></span><a href="domain_1.html#K6">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E1:74_1"><i><font color="Green">E53</font></i></a>, </i>;<br/><a NAME="E6:74_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E7:74_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">C1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i><a class="txt" href="chain_1.html#E2:74_1"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="chain_1.html#E3:74_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="chain_1.html#E1:74_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E4:74_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>()]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="ref" href="chain_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="chain_1.html#T5" target="_self">Th5</a>);</i><br/>


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