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

<b>set </b><font color="Maroon" title="c5">D</font> = <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> implies  ex <font color="Olive" title="b1">il</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">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <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"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">il</font></span>)</span></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) );<br/>
<a NAME="E1:70_1_1"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">m</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="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:70_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:70_1_1_1"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:70_1_1_1"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</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_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_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="E1:70_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> or <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">il</font> = <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</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="txt" href="amistd_2.html#E2:70_1_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="amistd_2.html#T20" target="_self" title="AMISTD_2:th.20">Th20</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">x</font> =  <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"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c7">il</font></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c8">x</font>
;<br/><b>thus </b><a NAME="E4:70_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c8">x</font>]
 <b>by </b><i><a class="txt" href="amistd_2.html#E1:70_1_1_1_1_1"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_2"/><i><font color="Green" title="E38">A5</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><a NAME="E2:70_1_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">x</font> = <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="amistd_2.html#E3:70_1_1_1"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">x</font>
;<br/><b>thus </b><a NAME="E4:70_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c7">x</font>]
 <b>by </b><i><a class="txt" href="amistd_2.html#E1:70_1_1_1_1_2"><i><font color="Green" title="E38">A5</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="c6">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:70_1_1"/><i><font color="Green" title="E37">A7</font></i>: 
for <font color="Olive" title="b1">k</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="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="amistd_2.html#E1:70_1_1"><i><font color="Green" title="E35">A1</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c7">f</font> =  <a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p2">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:70_1_1"/><i><font color="Green" title="E38">A8</font></i>: 
 <a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span>
 
;<br/>
<a NAME="E6:70_1_1"/><b>then </b><i><font color="Green" title="E39">A9</font></i>: 
 <a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_2.html#T10" target="_self" title="AMISTD_2:th.10">Th10</a></i>;<br/>
<i><font color="Green" title="E40">A10</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span>
<b>by </b><i><a class="txt" href="amistd_2.html#E3:70_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="pralg_2.html#K1" title="PRALG_2:func.1">DOM</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p2">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="amistd_2.html#E5:70_1_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="pralg_2.html#D2" title="PRALG_2:def.2">PRALG_2:def 2</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="amistd_2.html#T7" target="_self" title="AMISTD_2:th.7">Th7</a></i>
;<br/>
<a NAME="E8:70_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> holds <br/><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_1_3"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 ;<br/>

<a NAME="E2:70_1_1_3"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p2">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>,<font color="Maroon" title="c8">z</font>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E5:70_1_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="amistd_2.html#E7:70_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="amistd_2.html#D1" target="_self" title="AMISTD_2:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">z</font> as    <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>by </b><i><a class="txt" href="amistd_2.html#E1:70_1_1_3"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_3_1"/>
( <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> or <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_3_1_1"/><i><font color="Green" title="E43">A13</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><a NAME="E2:70_1_1_3_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">il</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">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</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"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">il</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="amistd_2.html#E3:70_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="amistd_2.html#E4:70_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="amistd_2.html#E1:70_1_1_3"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:70_1_1_3_1_1"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E1:70_1_1_3_1_1"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><a NAME="E2:70_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E3:70_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="amistd_2.html#E4:70_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="amistd_2.html#E1:70_1_1_3"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:70_1_1_3_1_2"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:70_1_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="amistd_2.html#E2:70_1_1_3"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:70_1_1"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p1">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E6:70_1_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="amistd_2.html#E7:70_1_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">IT</font> being   <a href="ami_1.html#NM3" title="AMI_1:NM.3">Instruction</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E11:70_1_1"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c8">IT</font>
 <b>and </b><br/><a NAME="E12:70_1_1"/><i><font color="Green" title="E42">A15</font></i>: 
 <a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c8">IT</font>
 ;<br/>
<b>take </b>
<font color="Maroon" title="c8">IT</font>
;<br/>

<b>thus </b><a NAME="E13:70_1_1"/>
 <a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c8">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E12:70_1_1"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:70_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c8">IT</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E3:70_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="amistd_2.html#E11:70_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E15:70_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span>
 ;<br/>

<a NAME="E16:70_1_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:70_1_1"/>
( ( <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> implies  ex <font color="Olive" title="b1">f</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">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> &amp; <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c8">IT</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</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"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> ) ) &amp; ( <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5" title="AMISTD_2:func.5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K19" title="AMI_1:func.19">InsCode</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> implies <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c8">IT</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressPart</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> ) )
 <b>by </b><i><a class="txt" href="amistd_2.html#E4:70_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="amistd_2.html#E11:70_1_1"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/>


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