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







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
 <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <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">J</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:20_1"/>
<font color="Maroon">z</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">J</font>
 ;<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="E3:20_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</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">z</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="E4:20_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</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#T1" target="_self">Th1</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <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">J</font>
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:20_1_1_2"/><b>then </b>
<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:20_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <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">J</font>
 <b>by </b><i><a class="ref" href="hallmar1.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">z</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="E3:20"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <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">J</font>
 ;<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="E5:20"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</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; <font color="Maroon">z</font> <a href="hidden.html#R2">in</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:20"/><i><font color="Green">E28</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T1" target="_self">Th1</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1"><b>suppose </b></a><a NAME="E1:20_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_2_1"/>
<font color="Maroon">z</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">J</font>
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_2"><b>suppose </b></a><a NAME="E1:20_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:20_2_2"/><b>then </b>
<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</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#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E3:20_2_2"/>
<font color="Maroon">z</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">J</font>
 <b>by </b><i><a class="ref" href="hallmar1.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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