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<div class="add">

<b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <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  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F</font>;<br/>



<b>thus </b><a NAME="E1:47"/>
(  ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">X</font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">A</font> implies <font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:47"/>
( <font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a> implies  ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">X</font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<b>then consider </b><font color="Maroon">G</font> being    <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">G</font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i/>;<br/>
<a NAME="E4:47_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i><a class="ref" href="hallmar1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E5:47_1"/>
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">G</font> holds <br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:47_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:47_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:47_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<b>hence </b><a NAME="E3:47_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:47_1"/><i><font color="Green">E28</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">G</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:47_1"/>
<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>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:47_1"/>
 ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">X</font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">A</font>
 ;<br/>


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


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