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<b>let </b><font color="Maroon" title="c1">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font> holds <br/> (  ex <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b2">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Olive" title="b1">A</font> iff <font color="Olive" title="b1">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c2">A</font> iff <font color="Maroon" title="c2">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> )</span><br/>



<b>thus </b><a NAME="E1:47"/>
(  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c2">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> )
 <b>by </b><i><a class="ref" href="hallmar1.html#T18" target="_self" title="HALLMAR1:th.18">Th18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> implies  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>thus </b><a NAME="E2:47"/>
( <font color="Maroon" title="c2">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a> implies  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c2">A</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E37">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c3">G</font> being    <a href="hallmar1.html#M4" title="HALLMAR1:mode.4">Singlification</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="hallmar1.html#V1" title="HALLMAR1:attr.1">Hall</a>
 <b>by </b><i><a class="ref" href="hallmar1.html#T31" target="_self" title="HALLMAR1:th.31">Th31</a></i>;<br/>
<a NAME="E4:47_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E1:47_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="hallmar1.html#T15" target="_self" title="HALLMAR1:th.15">Th15</a></i>;<br/>
<a NAME="E5:47_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G</font> holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G</font> implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E1:47_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="hallmar1.html#D6" target="_self" title="HALLMAR1:def.6">Def6</a></i>;<br/>
<b>assume </b><a NAME="E2:47_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<b>hence </b><a NAME="E3:47_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="hallmar1.html#E4:47_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="hallmar1.html#E1:47_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="hallmar1.html#D8" target="_self" title="HALLMAR1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:47_1"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E3:47_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="hallmar1.html#T17" target="_self" title="HALLMAR1:th.17">Th17</a></i>;<br/>
<b>thus </b><a NAME="E8:47_1"/>
 ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">X</font> <a href="hallmar1.html#R1" title="HALLMAR1:pred.1">is_a_system_of_different_representatives_of</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="hallmar1.html#E7:47_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="hallmar1.html#T28" target="_self" title="HALLMAR1:th.28">Th28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
