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

<b>let </b><font color="Maroon">R</font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="scmring2.html#M1">Data-Location</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">i1</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<i><font color="Green">E50</font></i>:  <a href="ami_5.html#K1">InsCode</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span> = 
 <a href="ami_5.html#K1">InsCode</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i><a class="ref" href="amistd_2.html#D14">AMISTD_2:def 14</a></i>
<br/>.= 
7
<b>by </b><i/>
<br/>.= 
 <a href="ami_5.html#K1">InsCode</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:110"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_2.html#D14">AMISTD_2:def 14</a></i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> 2
<b>by </b><i><a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E4:110"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:110_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:110_3"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="scmring3.html#T18" target="_self">Th18</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_3_1_1"><b>suppose </b></a><a NAME="E1:110_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:110_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_2.html#K1">PA</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_5.html#K1">InsCode</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="ami_1.html#NM2">Instruction-Location</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E4:110_3_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:110_3_1_1"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmring3.html#T17" target="_self">Th17</a>, <a class="ref" href="amistd_2.html#D14">AMISTD_2:def 14</a></i>;<br/><font color="Maroon">f</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="scmring3.html#T20" target="_self">Th20</a>, </i>
<br/>.= 
<font color="Maroon">i1</font>
<b>by </b><i><a class="ref" href="scmring3.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="scmring3.html#T19" target="_self">Th19</a>, <a class="ref" href="scmring3.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_3_1_2"><b>suppose </b></a><a NAME="E1:110_3_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:110_3_1_2"/>
 <a href="ami_2.html#K2">SCM-Data-Loc</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:110_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_2.html#K1">PA</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K5">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_5.html#K1">InsCode</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="scmring3.html#T19" target="_self">Th19</a>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="scmring3.html#T17" target="_self">Th17</a>, <a class="ref" href="amistd_2.html#D14">AMISTD_2:def 14</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="scmring3.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="scmring3.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="amistd_2.html#K4">AddressPart</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p4">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:110"/><b>then </b>
 <a href="amistd_2.html#K4">AddressPart</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K8">IncAddr</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="amistd_2.html#K4">AddressPart</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p3">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmring3.html#T15" target="_self">Th15</a>, <a class="ref" href="scmring3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:110"/>
 <a href="amistd_2.html#K8">IncAddr</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="scmring2.html#K10">=0_goto</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmring3.html#T14" target="_self">Th14</a>, <a class="ref" href="amistd_2.html#T16">AMISTD_2:16</a></i>;<br/>


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