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

<b>let </b><font color="Maroon">F</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F</font>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:44"/><i><font color="Green">E24</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E3:44"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ai</font> = <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  non <a href="realset1.html#V1">trivial</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pencil_1.html#T4">PENCIL_1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:44"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font>
 <b>and </b><br/><a NAME="E7:44"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font>
 <b>and </b><br/><a NAME="E8:44"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="realset1.html#T14">REALSET1:14</a></i>;<br/>
<a NAME="E9:44"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:44_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E10:44"/><i><font color="Green">E30</font></i>: 
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</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/> not  <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Olive">z</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<a NAME="E11:44"/><b>then </b>
not  <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T2" target="_self">Th2</a></i>;<br/>
<b>then consider </b><font color="Maroon">JJ1</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:44"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">JJ1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">JJ1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span>,<font color="Maroon">JJ1</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E14:44"/>
 ex <font color="Olive">J1</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">J1</font> &amp; <font color="Olive">J1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">J1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Olive">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ1</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1"><b>suppose </b></a><a NAME="E1:44_2_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ1</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">J1</font> = <font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>;<br/><i><font color="Green">E72</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">JJ1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="euler_1.html#T5">EULER_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">JJ1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/><a NAME="E3:44_2_1_1"/>
 <a href="hallmar1.html#K1" target="_self">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span>,<font color="Maroon">JJ1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="hallmar1.html#D1" target="_self">Def1</a>, , </i>;<br/><a NAME="E4:44_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hallmar1.html#D1" target="_self">Def1</a>, <a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, <a class="ref" href="sppol_1.html#T6">SPPOL_1:6</a></i>;<br/><a NAME="E5:44_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="hallmar1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:44_2_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E7:44_2_1_1"/><b>then </b>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T54">ZFMISC_1:54</a></i>;<br/><b>hence </b><a NAME="E8:44_2_1_1"/>
 ex <font color="Olive">J1</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">J1</font> &amp; <font color="Olive">J1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">J1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Olive">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_2"><b>suppose </b></a><a NAME="E1:44_2_1_2"/><i><font color="Green">E75</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ1</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">J1</font> = <font color="Maroon">JJ1</font>;<br/><b>take </b>
<font color="Maroon">JJ1</font>
;<br/><a NAME="E2:44_2_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">JJ1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hallmar1.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E3:44_2_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">JJ1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">JJ1</font></span>)</span>
 <b>by </b><i><a class="ref" href="hallmar1.html#D1" target="_self">Def1</a>, , </i>;<br/><b>hence </b><a NAME="E4:44_2_1_2"/>
( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ1</font> &amp; <font color="Maroon">JJ1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">JJ1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">JJ1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">J1</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:44"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font>
 <b>and </b><br/><a NAME="E17:44"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">J1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">J1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 ;<br/>
<b>set </b><font color="Maroon">S1</font> = <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<a NAME="E18:44"/>
not  <a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">JJ2</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:44"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">JJ2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">JJ2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">JJ2</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E21:44"/>
 ex <font color="Olive">J2</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">J2</font> &amp; <font color="Olive">J2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">J2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Olive">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_3_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ2</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1_1"><b>suppose </b></a><a NAME="E1:44_3_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">J2</font> = <font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>;<br/><i><font color="Green">E81</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">JJ2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="euler_1.html#T5">EULER_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">JJ2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/><a NAME="E3:44_3_1_1"/>
 <a href="hallmar1.html#K1" target="_self">union</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">JJ2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:44_3_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T5" target="_self">Th5</a>, <a class="ref" href="sppol_1.html#T6">SPPOL_1:6</a></i>;<br/><a NAME="E5:44_3_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:44_3_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E7:44_3_1_1"/><b>then </b>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ2</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T54">ZFMISC_1:54</a></i>;<br/><b>hence </b><a NAME="E8:44_3_1_1"/>
 ex <font color="Olive">J2</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">J2</font> &amp; <font color="Olive">J2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">J2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Olive">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T6" target="_self">Th6</a>, <a class="ref" href="hallmar1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1_2"><b>suppose </b></a><a NAME="E1:44_3_1_2"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">J2</font> = <font color="Maroon">JJ2</font>;<br/><b>take </b>
<font color="Maroon">JJ2</font>
;<br/><a NAME="E2:44_3_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">JJ2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:44_3_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">JJ2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">JJ2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:44_3_1_2"/>
( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JJ2</font> &amp; <font color="Maroon">JJ2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">JJ2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">JJ2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">J2</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:44"/><i><font color="Green">E86</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font>
 <b>and </b><br/><a NAME="E24:44"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">J2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K2" target="_self">Cut</a> <font color="Maroon">A</font>,<font color="Maroon">i</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">J2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> )
 ;<br/>
<b>set </b><font color="Maroon">S2</font> = <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<a NAME="E25:44"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, <a class="ref" href="hallmar1.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E26:44"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E27:44"/>
<span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Ai</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_4"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Ai</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_4_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font> )
 </span><b>by </b><i><a class="ref" href="hallmar1.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_4_1_1"><b>suppose </b></a><a NAME="E1:44_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_4_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><a NAME="E4:44_4_1_1"/>
( ( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> or <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> ) &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:44_4_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 <b>by </b><i/>;<br/><a NAME="E6:44_4_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:44_4_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:44_4_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E9:44_4_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E10:44_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_4_1_2"><b>suppose </b></a><a NAME="E1:44_4_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font>
 ;<br/><div class="add"><a NAME="E2:44_4_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:44_4_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/><a NAME="E5:44_4_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E6:44_4_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:44_4_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T2" target="_self">Th2</a>, , , </i>;<br/><a NAME="E8:44_4_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E9:44_4_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:44_4_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E11:44_4_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E12:44_4_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><b>hence </b><a NAME="E13:44_4_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E28:44"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Ai</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E29:44"/>
<span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span>
 ;<br/>

<a NAME="E2:44_5"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:44_5"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:44_5"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E7:44_5"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</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> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:44_5"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:44_5"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:44_5"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</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="E11:44_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hallmar1.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:44"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E31:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p4">(<span class="default"><font color="Maroon">Ai</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Ai</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E32:44"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Ai</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:44"/>
 <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_6"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:44_6"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:44_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:44_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:44"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E35:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Ai</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E36:44"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J12</font> = <font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E38:44"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E39:44"/>
<font color="Maroon">J1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E40:44"/><b>then </b>
<font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E41:44"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E42:44"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">J1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">J2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="hallmar1.html#T3" target="_self">Th3</a>, <a class="ref" href="hallmar1.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E43:44"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E44:44"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E46:44"/>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E47:44"/><b>then </b>
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E48:44"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E49:44"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E50:44"/><i><font color="Green">E133</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E51:44"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/>
<a NAME="E52:44"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">Ai</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#D2" target="_self">Def2</a>, , <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>
<a NAME="E53:44"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">Ai</font> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T10" target="_self">Th10</a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E54:44"/>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</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">J1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E55:44"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">J2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">J1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">J2</font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E56:44"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div>
