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<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="ami_1.html#V2" title="AMI_1:attr.2">void</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ami_3.html#V1" title="AMI_3:attr.1">programmed</a>  <a href="ami_1.html#M1" title="AMI_1:mode.1">FinPartState</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">M</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> such that </b><br/><a NAME="E3:38"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c5">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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</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> 
 <b>and </b><br/><a NAME="E4:38"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c3">F</font> <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="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c5">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">N</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> such that </b><br/><a NAME="E6:38"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c6">N</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</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">G</font> </span>}</span> 
 <b>and </b><br/><a NAME="E7:38"/><i><font color="Green" title="E24">A5</font></i>: 
 <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c4">G</font> <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="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c6">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_3.html#D2" target="_self" title="AMISTD_3:def.2">Def2</a></i>;<br/>
<a NAME="E8:38"/>
<font color="Maroon" title="c5">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</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="E1:38_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
 ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c7">a</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="c3">F</font> )
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:38"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E3:38_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:38"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E6:38"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="amistd_3.html#E2:38_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:38"/><b>then </b>
 <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c6">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c5">M</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T19" title="CQC_SIM1:th.19">CQC_SIM1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:38"/>
 <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c4">G</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a>  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E4:38"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="amistd_3.html#E7:38"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="amistd_1.html#T28" title="AMISTD_1:th.28">AMISTD_1:28</a></i>;<br/>


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