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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="amistd_1.html#K7" target="_self">locnum</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<a NAME="E1:84_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E2:84_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>;<br/>
<a NAME="E4:84_1_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E6:84_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="amistd_1.html#K7" target="_self">locnum</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 
;<br/>
<a NAME="E7:84_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</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="E1:84_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<a NAME="E2:84_1_1_1"/><b>then </b>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K7" target="_self">locnum</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:84_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">l</font>) where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<b>take </b>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">M</font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E9:84_1_1"/>
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K7" target="_self">locnum</a> <font color="Olive">l</font></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> : <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span>  &amp;  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">M</font></span>)</span> )
 ;<br/>


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