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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> $1;<br/>
<b>reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ami_1.html#V15" title="AMI_1:attr.15">programmed</a> <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="amistd_3.html#E1:35"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="amistd_3.html#E2:35"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span> ;<br/>
<a NAME="E2:35_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E3:35_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="ami_1.html#S1" title="AMI_1:sch.1">AMI_1:sch 1</a>(<a class="txt" href="amistd_3.html#E2:35_1_1"><i><font color="Green" title="E19">A3</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c6">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font>;<br/>
<a NAME="E5:35_1_1"/>
( <font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="ami_1.html#D40" title="AMI_1:def.40">AMI_1:def 40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c6">l</font> as    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/>
<a NAME="E7:35_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span> 
 
;<br/>
<a NAME="E8:35_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span> 
 ;<br/>

<a NAME="E2:35_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">l</font> being   <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:35_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">l</font>) where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span>  as  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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="amistd_3.html#E3:35_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="amistd_3.html#E7:35_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>
<b>take </b>
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c8">M</font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">M</font>
;<br/>

<b>thus </b><a NAME="E10:35_1_1"/>
( <font color="Maroon" title="c8">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> </span>}</span>  &amp;  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c8">M</font></span>)</span> )
 ;<br/>


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