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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:47"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T18" target="_self">Th18</a></i>;<br/>

<b>thus </b><a NAME="E2:47"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">c<sub>2</sub></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">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hallmar1.html#M4" target="_self">Singlification</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 <b>by </b><i><a class="ref" href="hallmar1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:47_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:47_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="hallmar1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E5:47_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:47_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self">Def6</a></i>;<br/>
<b>assume </b><a NAME="E2:47_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:47_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hallmar1.html#E1:47_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hallmar1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:47_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:47_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="hallmar1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E8:47_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E7:47_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hallmar1.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E9:47_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hallmar1.html#R1" target="_self">is_a_system_of_different_representatives_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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


</div>
