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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Si</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">FSi</font> be    <a href="prob_3.html#M1" target="_self">FinSequence</a> of <font color="Maroon">Si</font>;<br/>





<b>consider </b><font color="Maroon">A1</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:57"/><i><font color="Green">E36</font></i>: 
( ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font> holds <br/><font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSi</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Olive">k</font> ) &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st not <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font> holds <br/><font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i/>;<br/>
<a NAME="E3:57"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1"/>
(  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font> or <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font>
 ;<br/><div class="add"><a NAME="E2:57_1_1_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:57_1_1_1"/>
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T42">PROB_1:42</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font>
 ;<br/><div class="add"><a NAME="E2:57_1_1_2"/><b>then </b>
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSi</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:57_1_1_2"/>
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T2" target="_self">Th2</a>, <a class="txt" href="prob_3.html#E1"><i><font color="Green">Lemma47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A1</font> as    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Si</font> <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">A1</font>
;<br/>

<b>thus </b><a NAME="E5:57"/>
( ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font> holds <br/><font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSi</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Olive">k</font> ) &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st not <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSi</font> holds <br/><font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i/>;<br/>


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