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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>given </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:27"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:27"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="hallmar1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:27"/>
for <font color="Olive">J</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">J</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> holds <br/> <a href="card_1.html#K4">card</a> <font color="Olive">J</font> <a href="xxreal_0.html#R1">&lt;=</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="Olive">J</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">J</font>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font>;<br/>
<a NAME="E2:27_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E3:27_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hallmar1.html#E6:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:27_1"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E5:27_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">A</font>,<font color="Maroon">J</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><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="E1:27_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:27_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E2:27_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="hallmar1.html#E3:27_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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">a</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="hallmar1.html#E4:27"><i><font color="Green">E24</font></i></a>, <a class="txt" href="hallmar1.html#E6:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="hallmar1.html#E1:27_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="hallmar1.html#E3:27_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E7:27_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <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">a</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E3:27_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1"/>
<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">J</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E6:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="hallmar1.html#E1:27_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="hallmar1.html#E3:27_1"><i><font color="Green">E30</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:27_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">J</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">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">J</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="hallmar1.html#E1:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:27_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">J</font> <a href="xxreal_0.html#R1">&lt;=</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">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:27"/>
<font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i/>;<br/>


</div>
