<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:140_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_1_1"><b>suppose </b></a><a NAME="E1:140_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:140_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Olive">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Olive">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) )
 ;<br/><b>reconsider </b><font color="Maroon">E</font> =  <a href="xboole_0.html#K1">{}</a>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b>
<font color="Maroon">E</font>
;<br/><b>thus </b><a NAME="E4:140_1_1"/>
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="140_1_2"><b>suppose </b></a><a NAME="E1:140_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">D</font>] <b>means </b>( ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Olive">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Olive">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) ) ) implies  ex <font color="Olive">C</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> ) );<br/><a NAME="E3:140_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">D</font>]
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><a NAME="E4:140_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">D</font><br/> for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] &amp; not <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">s</font></span><a href="setwiseo.html#K2">.}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:140_1_2_1"/><i><font color="Green">E49</font></i>: 
( ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/> ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Olive">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Olive">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) ) ) implies  ex <font color="Olive">C</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">S</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:140_1_2_1"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:140_1_2_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> holds <br/> ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Olive">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Olive">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_2_1_1"><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="E1:140_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:140_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:140_1_2_1_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T55" target="_self">Th55</a>, <a class="ref" href="group_2.html#T59" target="_self">Th59</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:140_1_2_1_1"/><b>then </b>
 ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Maroon">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) )
 <b>by </b><i><a class="ref" href="group_2.html#T57" target="_self">Th57</a></i>;<br/><b>hence </b><a NAME="E5:140_1_2_1_1"/>
<font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D1</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">S</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>
<a NAME="E6:140_1_2_1"/><i><font color="Green">E53</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E7:140_1_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:140_1_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:140_1_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Maroon">s</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">s</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) )
 <b>by </b><i><a class="ref" href="group_2.html#T57" target="_self">Th57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">s</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="group_2.html#T60" target="_self">Th60</a></i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:140_1_2_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">S</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tarski.html#K3">union</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:140_1_2_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:140_1_2_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:140_1_2_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:140_1_2_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E9:140_1_2_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E10:140_1_2_1_2"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="group_2.html#T55" target="_self">Th55</a>, , <a class="ref" href="group_2.html#T62" target="_self">Th62</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:140_1_2_1_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Maroon">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) )
 <b>by </b><i><a class="ref" href="group_2.html#T57" target="_self">Th57</a></i>;<br/><a NAME="E13:140_1_2_1_2"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="group_2.html#T63" target="_self">Th63</a>, <a class="ref" href="group_2.html#T64" target="_self">Th64</a></i>;<br/><b>hence </b><a NAME="E14:140_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="group_2.html#T61" target="_self">Th61</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:140_1_2_1"/><i><font color="Green">E135</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59" target="_self">Th59</a>, <a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">D2</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">D2</font>
;<br/>

<b>thus </b><a NAME="E15:140_1_2_1"/>
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<div><i><font color="Green">E136</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_2_1_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>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:140_1_2_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:140_1_2_1_3"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:140_1_2_1_3"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Maroon">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) )
 <b>by </b><i><a class="ref" href="group_2.html#T57" target="_self">Th57</a></i>;<br/><b>take </b><font color="Maroon">B</font> = <font color="Maroon">B</font>;<br/><b>thus </b><a NAME="E5:140_1_2_1_3"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E6:140_1_2_1_3"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E7:140_1_2_1_3"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E8:140_1_2_1_3"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:140_1_2_1_3"/>
( <font color="Maroon">Y</font>,<font color="Maroon">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="group_2.html#T62" target="_self">Th62</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:140_1_2_1"/><i><font color="Green">E179</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="group_2.html#T55" target="_self">Th55</a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<b>thus </b> <a href="card_1.html#K4">card</a> <font color="Maroon">D2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="group_2.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T60" target="_self">Th60</a>, , , <a class="ref" href="group_2.html#T61" target="_self">Th61</a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><a NAME="E5:140_1_2"/><i><font color="Green">E180</font></i>: 
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2">SETWISEO:sch 2</a>(, <a class="ref" href="group_2.html#T53" target="_self">Th53</a>);<br/></i><a NAME="E6:140_1_2"/>
<font color="Maroon">X</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:140_1_2"/>
( ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Z</font> holds <br/>( <font color="Olive">Y</font>,<font color="Olive">Z</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Olive">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Z</font> ) ) ) ) implies  ex <font color="Olive">C</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X</font></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
