<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c4">G</font> =  <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1_1_1"/>
(  <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not  <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1"><b>suppose </b></a><a NAME="E1:117_1_1_1_1"/>
 <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">H</font> =  <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#M1" title="AMI_1:mode.1">FinPartState</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E3:117_1_1_1_1"/>
<font color="Maroon" title="c5">H</font> is <a href="amistd_1.html#V9" title="AMISTD_1:attr.9">lower</a>
 ;<br/><b>hence </b><a NAME="E4:117_1_1_1_1"/>
 <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> is <a href="amistd_1.html#V9" title="AMISTD_1:attr.9">lower</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_2"><b>suppose </b></a><a NAME="E1:117_1_1_1_2"/>
not  <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <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#M1" title="AMI_1:mode.1">FinPartState</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E3:117_1_1_1_2"/>
<font color="Maroon" title="c5">G</font> is <a href="amistd_1.html#V9" title="AMISTD_1:attr.9">lower</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">l</font> be   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction-Location</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_1.html#D20" title="AMISTD_1:def.20">AMISTD_1:def 20</a><br/>

<b>assume </b><a NAME="E1:117_1_1_1_2_1"/><i><font color="Green" title="E64">A1</font></i>: 
<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="c5">G</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">m</font> be   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction-Location</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E2:117_1_1_1_2_1"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c6">l</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">M</font> being  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="membered.html#V5" title="MEMBERED:attr.5">natural-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:117_1_1_1_2_1"/><i><font color="Green" title="E66">A3</font></i>: 
<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">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">w</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="E5:117_1_1_1_2_1"/><i><font color="Green" title="E67">A4</font></i>: 
 <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</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="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_1.html#D21" title="AMISTD_1:def.21">AMISTD_1:def 21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">R</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T4" title="RELAT_1:th.4">RELAT_1:4</a></i>;<br/>
<a NAME="E7:117_1_1_1_2_1"/>
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T45" title="CQC_LANG:th.45">CQC_LANG:45</a></i>;<br/>
<a NAME="E8:117_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E9:117_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E69">A6</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="amistd_2.html#T47" target="_self" title="AMISTD_2:th.47">Th47</a></i>;<br/>
<a NAME="E10:117_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E70">A7</font></i>: 
<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="c3">F</font>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E1:117_1_1_1_2_1"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/>
<a NAME="E11:117_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E71">A8</font></i>: 
<font color="Maroon" title="c7">m</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_2.html#E2:117_1_1_1_2_1"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="amistd_1.html#D20" title="AMISTD_1:def.20">AMISTD_1:def 20</a></i>;<br/>
<a NAME="E12:117_1_1_1_2_1"/>
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E4:117_1_1_1_2_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="amistd_2.html#E10:117_1_1_1_2_1"><i><font color="Green" title="E70">A7</font></i></a></i>;<br/>
<a NAME="E13:117_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E72">A9</font></i>: 
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_1_1_1_2_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font>
 ;<br/><a NAME="E2:117_1_1_1_2_1_1"/><b>then </b>
 <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</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="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E2:117_1_1_1_2_1"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></i>;<br/><a NAME="E3:117_1_1_1_2_1_1"/><b>then </b>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:117_1_1_1_2_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="amistd_1.html#T28" title="AMISTD_1:th.28">AMISTD_1:28</a></i>;<br/><a NAME="E4:117_1_1_1_2_1_1"/><b>then </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="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</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="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E13:117_1_1_1_2_1"><i><font color="Green" title="E72">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:117_1_1_1_2_1_1"/><b>then </b>
 <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</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="c5">G</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E1:117_1_1_1_2_1"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="amistd_2.html#E5:117_1_1_1_2_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></i>;<br/><a NAME="E6:117_1_1_1_2_1_1"/><b>then </b>
not  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E8:117_1_1_1_2_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="amistd_2.html#E9:117_1_1_1_2_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:117_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:117_1_1_1_2_1"/><b>then </b>
not <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:117_1_1_1_2_1"/>
<font color="Maroon" title="c7">m</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="c5">G</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E8:117_1_1_1_2_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="amistd_2.html#E9:117_1_1_1_2_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="amistd_2.html#E11:117_1_1_1_2_1"><i><font color="Green" title="E71">A8</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:117_1_1_1_2"/>
 <a href="amistd_2.html#K11" title="AMISTD_2:func.11">CutLastLoc</a> <font color="Maroon" title="c3">F</font> is <a href="amistd_1.html#V9" title="AMISTD_1:attr.9">lower</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
