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

<b>set </b><font color="Maroon">G</font> =  <a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">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" target="_self">CutLastLoc</a> <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font> is <a href="xboole_0.html#V1">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" target="_self">CutLastLoc</a> <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">H</font> =  <a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font> as  <a href="xboole_0.html#V1">empty</a>  <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">S</font> ;<br/><a NAME="E3:117_1_1_1_1"/>
<font color="Maroon">H</font> is <a href="amistd_1.html#V9">lower</a>
 ;<br/><b>hence </b><a NAME="E4:117_1_1_1_1"/>
 <a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font> is <a href="amistd_1.html#V9">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" target="_self">CutLastLoc</a> <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">G</font> =  <a href="amistd_2.html#K11" target="_self">CutLastLoc</a> <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">S</font> ;<br/><a NAME="E3:117_1_1_1_2"/>
<font color="Maroon">G</font> is <a href="amistd_1.html#V9">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">l</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_1.html#D20">AMISTD_1:def 20</a><br/>

<b>assume </b><a NAME="E1:117_1_1_1_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E2:117_1_1_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="amistd_1.html#R1">&lt;=</a> <font color="Maroon">l</font>
 ;<br/>

<b>consider </b><font color="Maroon">M</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V5">natural-membered</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:117_1_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<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">locnum</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</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">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 <b>and </b><br/><a NAME="E5:117_1_1_1_2_1"/><i><font color="Green">E57</font></i>: 
 <a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">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>
 <b>by </b><i><a class="ref" href="amistd_1.html#D21">AMISTD_1:def 21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T4">RELAT_1:4</a></i>;<br/>
<a NAME="E7:117_1_1_1_2_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_1.html#T19">AMI_1:19</a></i>;<br/>
<a NAME="E8:117_1_1_1_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E9:117_1_1_1_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E10:117_1_1_1_2_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E10:117_1_1_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E13:117_1_1_1_2_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E11:117_1_1_1_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="amistd_1.html#D20">AMISTD_1:def 20</a></i>;<br/>
<a NAME="E12:117_1_1_1_2_1"/>
 <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="amistd_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E13:117_1_1_1_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">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">m</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:117_1_1_1_2_1_1"/><b>then </b>
 <a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font> <a href="amistd_1.html#R1">&lt;=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="amistd_1.html#D13">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">max</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="amistd_1.html#T28">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">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">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>
 <b>by </b><i><a class="ref" href="amistd_2.html#T10" target="_self">Th10</a>, <a class="ref" href="xxreal_0.html#T1">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">LastLoc</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E13:117_1_1_1_2_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="amistd_1.html#D13">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">LastLoc</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="amistd_2.html#D2" target="_self">Def2</a>, , <a class="ref" href="xboole_0.html#D4">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">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">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K10">LastLoc</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E16:117_1_1_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="amistd_2.html#D2" target="_self">Def2</a>, , <a class="ref" href="amistd_2.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D4">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" target="_self">CutLastLoc</a> <font color="Maroon">F</font> is <a href="amistd_1.html#V9">lower</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
