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

<b>set </b><font color="Maroon" title="c4">D</font> = the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </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="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">l</font></span>)</span>,<font color="Olive" title="b1">l</font></span>)</span></span>)</span> implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">l</font></span>)</span>,<font color="Olive" title="b1">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 ) &amp; ( not $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">l</font></span>)</span>,<font color="Olive" title="b1">l</font></span>)</span></span>)</span> implies $3 <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>,0 ) );<br/>
<a NAME="E1:53_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">x</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><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">z</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <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>;<br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x</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/>
<b>set </b><font color="Maroon" title="c8">z</font> = <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1"/>
( <font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>take </b>
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
;<br/><a NAME="E3:53_1_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <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="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>] )
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:53_1_1_1_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="amistd_3.html#E2:53_1_1_1_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2"/><i><font color="Green" title="E27">A4</font></i>: 
not <font color="Maroon" title="c6">y</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> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">x</font></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>take </b>
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,0
;<br/><b>thus </b><a NAME="E2:53_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>, <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,0]
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:53_1_1_1_1_2"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
for <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><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">n</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="amistd_3.html#E1:53_1_1"><i><font color="Green" title="E26">A1</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </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="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">l</font></span>)</span>,<font color="Olive" title="b1">l</font></span>)</span> );<br/>
<a NAME="E4:53_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
for <font color="Olive" title="b1">d</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><br/> for <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">k2</font>,<font color="Olive" title="b1">d</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">k1</font>,<font color="Olive" title="b1">d</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">d</font> be    <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>;<br/><b>let </b><font color="Maroon" title="c7">k1</font>, <font color="Maroon" title="c8">k2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53_1_1_2"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c7">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k2</font>
 <b>and </b><br/><a NAME="E2:53_1_1_2"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k2</font>,<font color="Maroon" title="c6">d</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">d</font> = <font color="Maroon" title="c6">d</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/>
<b>take </b>
<font color="Maroon" title="c9">d</font>
;<br/>

<a NAME="E4:53_1_1_2"/>
<font color="Maroon" title="c7">k1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">k2</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:53_1_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="card_1.html#T56" title="CARD_1:th.56">CARD_1:56</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1_2"/>
( <font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c7">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">d</font></span>)</span>,<font color="Maroon" title="c9">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:53_1_1_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">T</font> being   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</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><b> such that </b><br/><a NAME="E6:53_1_1"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c6">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c3">M</font>
 <b>and </b><br/><a NAME="E7:53_1_1"/><i><font color="Green" title="E30">A10</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">T</font> holds <br/> (  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">k</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c6">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font>] </span>}</span>  &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Maroon" title="c6">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font>] holds <br/><font color="Maroon" title="c6">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b2">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span>,<font color="Olive" title="b2">n</font> ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S10" title="TREES_2:sch.10">TREES_2:sch 10</a>(<a class="txt" href="amistd_3.html#E4:53_1_1"><i><font color="Green" title="E28">A6</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">T</font> = <font color="Maroon" title="c6">T</font> as    <a href="ami_1.html#M5" title="AMI_1:mode.5">IL-DecoratedTree</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="ami_1.html#D37" title="AMI_1:def.37">AMI_1:def 37</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">T</font>
;<br/>

<a NAME="E9:53_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c7">T</font>
 
<b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<b>hence </b><a NAME="E10:53_1_1"/>
<font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">FirstLoc</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E6:53_1_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="ami_1.html#D38" title="AMI_1:def.38">AMI_1:def 38</a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">T</font>;<br/>

<a NAME="E11:53_1_1"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#D38" title="AMI_1:def.38">AMI_1:def 38</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c8">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">$1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of <font color="Maroon" title="c2">S</font>] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,$2</span>)</span>,$2</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">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="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">l</font>] ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c10">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font>] </span>}</span> ;<br/>
<a NAME="E12:53_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">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="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">l</font>] ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="ami_1.html#S2" title="AMI_1:sch.2">AMI_1:sch 2</a>();<br/></i>
<b>hence </b><a NAME="E13:53_1_1"/><i><font color="Green" title="E32">A12</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E7:53_1_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="amistd_3.html#E11:53_1_1"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E14:53_1_1"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font></span>)</span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E16:53_1_1"/><i><font color="Green" title="E34">A14</font></i>: 
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font>
 <b>and </b><br/><a NAME="E17:53_1_1"/><i><font color="Green" title="E35">A15</font></i>: 
( <font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c12">l</font></span>)</span>,<font color="Maroon" title="c12">l</font></span>)</span></span>)</span> implies <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font></span>)</span>,<font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c12">l</font></span>)</span>,<font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> )
 <b>and </b><br/><a NAME="E18:53_1_1"/>
( not <font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c12">l</font></span>)</span>,<font color="Maroon" title="c12">l</font></span>)</span></span>)</span> implies <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font></span>)</span>,<font color="Maroon" title="c11">m</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>,0 )
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:53_1_1"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/>
<a NAME="E19:53_1_1"/>
<font color="Maroon" title="c8">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <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">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="amistd_3.html#E14:53_1_1"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/>
<a NAME="E20:53_1_1"/><b>then </b>
<font color="Maroon" title="c7">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E13:53_1_1"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="ami_1.html#D38" title="AMI_1:def.38">AMI_1:def 38</a></i>;<br/>
<b>hence </b><a NAME="E21:53_1_1"/>
<font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">M</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c8">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E7:53_1_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="amistd_3.html#E11:53_1_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="amistd_3.html#E14:53_1_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="amistd_3.html#E16:53_1_1"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="amistd_3.html#E17:53_1_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>;<br/>


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