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

<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">i1</font> be    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of  <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c4">V</font> = <font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c3">i1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="scmring1.html#K1" title="SCMRING1:func.1">SCM-Instr</a> <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">i5</font> = <font color="Maroon" title="c3">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr-Loc</a>  <b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">s</font> being   <a href="scmring1.html#NM1" title="SCMRING1:NM.1">SCM-State</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">f</font> = the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E29">A1</font></i>: 
the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="scmring1.html#K2" title="SCMRING1:func.2">SCM-OK</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<a NAME="E5:37"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:37"/><b>then </b>
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E31">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
<br/>.= 
 <a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <font color="Maroon" title="c5">i5</font>
<b>by </b><i><a class="ref" href="scmring2.html#T11" target="_self" title="SCMRING2:th.11">Th11</a></i>
;<br/>
<a NAME="E8:37"/><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Memory</a> 
 
<b>by </b><i><a class="ref" href="ami_2.html#T30" title="AMI_2:th.30">AMI_2:30</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:37"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2" title="SCMRING1:func.2">SCM-OK</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<i><font color="Green" title="E34">A6</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Memory</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E9:37"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Memory</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Memory</a> 
<b>by </b><i><a class="txt" href="scmring2.html#E8:37"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E11:37"/><i><font color="Green" title="E35">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Memory</a> 
 
<b>by </b><i><a class="txt" href="scmring2.html#E4:37"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:37"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:37_3"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c9">x</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> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1"><b>suppose </b></a><a NAME="E1:37_3_1_1"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><a NAME="E2:37_3_1_1"/><b>then </b>
the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr-Loc</a> 
 <b>by </b><i><a class="txt" href="scmring2.html#E4:37"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="ami_2.html#T30" title="AMI_2:th.30">AMI_2:30</a>, <a class="ref" href="scmring1.html#T2" title="SCMRING1:th.2">SCMRING1:2</a></i>;<br/><b>hence </b><a NAME="E3:37_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="scmring2.html#E7:37"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="scmring2.html#E1:37_3_1_1"><i><font color="Green" title="E37">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_2"><b>suppose </b></a><a NAME="E1:37_3_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><div class="add"><a NAME="E2:37_3_1_2"/><b>then </b>
not <font color="Maroon" title="c9">x</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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmring2.html#E5:37"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:37_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:37_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="scmring2.html#E4:37"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="scmring2.html#E1:37_3"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c6">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span> as   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="scmring2.html#E10:37"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="scmring2.html#E11:37"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c9">t</font> as   <a href="scmring1.html#NM1" title="SCMRING1:NM.1">SCM-State</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<a NAME="E15:37"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E16:37"/><b>then </b>
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">i1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E36">A10</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c3">i1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<a NAME="E18:37"/><i><font color="Green" title="E37">A11</font></i>: 
7 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> 8
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E19:37"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c2">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="scmring2.html#T1" target="_self" title="SCMRING2:th.1">Th1</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green" title="E39">A13</font></i>: 
the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr-Loc</a> 
 
<b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E21:37"/><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c3">i1</font> is <a href="ami_1.html#V3" title="AMI_1:attr.3">halting</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5"/>
( <font color="Maroon" title="c10">w</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c10">w</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1"><b>suppose </b></a><a NAME="E1:37_5_1"/><i><font color="Green" title="E41">A15</font></i>: 
<font color="Maroon" title="c10">w</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:37_5_1"/><i><font color="Green" title="E42">A16</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c10">w</font>
 <b>by </b><i><a class="ref" href="scmring2.html#T10" target="_self" title="SCMRING2:th.10">Th10</a></i>;<br/><a NAME="E3:37_5_1"/><i><font color="Green" title="E43">A17</font></i>: 
 <a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E4:37_5_1"/><i><font color="Green" title="E44">A18</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c3">i1</font></span>)</span>,<font color="Maroon" title="c9">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="scmring2.html#E21:37"><i><font color="Green" title="E40">A14</font></i></a>, <a class="txt" href="scmring2.html#E2:37_5_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="ami_1.html#D8" title="AMI_1:def.8">AMI_1:def 8</a></i>;<br/><a NAME="E5:37_5_1"/>
<font color="Maroon" title="c10">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring2.html#E3:37_5_1"><i><font color="Green" title="E43">A17</font></i></a>, <a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of  <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</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/><a NAME="E7:37_5_1"/>
<font color="Maroon" title="c2">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="scmring2.html#T1" target="_self" title="SCMRING2:th.1">Th1</a></i>;<br/><a NAME="E8:37_5_1"/><b>then </b>
<font color="Maroon" title="c9">t</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="scmring2.html#E18:37"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="scmring2.html#E20:37"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="scmring2.html#E1:37_5_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="scmring1.html#T19" title="SCMRING1:th.19">SCMRING1:19</a></i>;<br/><a NAME="E9:37_5_1"/><b>then </b><i><font color="Green" title="E45">A19</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c3">i1</font></span>)</span>,<font color="Maroon" title="c9">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c11">e</font>
 <b>by </b><i><a class="txt" href="scmring2.html#E2:37_5_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="scmring2.html#T18" target="_self" title="SCMRING2:th.18">Th18</a></i>;<br/><a NAME="E10:37_5_1"/>
 ex <font color="Olive" title="b1">mj</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr-Loc</a>  st <br/>( <font color="Olive" title="b1">mj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font> &amp;  <a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <font color="Olive" title="b1">mj</font> )
 <b>by </b><i><a class="ref" href="scmring2.html#D10" target="_self" title="SCMRING2:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E11:37_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmring2.html#E4:37_5_1"><i><font color="Green" title="E44">A18</font></i></a>, <a class="txt" href="scmring2.html#E9:37_5_1"><i><font color="Green" title="E45">A19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_2"><b>suppose </b></a><a NAME="E1:37_5_2"/><i><font color="Green" title="E41">A20</font></i>: 
<font color="Maroon" title="c10">w</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><font color="Maroon" title="c10">w</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c3">i1</font></span>)</span> = 
 <a href="scmring1.html#K4" title="SCMRING1:func.4">SCM-Chg</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c5">i5</font>

<br/>.= 
 <a href="scmring1.html#K4" title="SCMRING1:func.4">SCM-Chg</a> <font color="Maroon" title="c10">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K10" title="SCMRING1:func.10">cjump_address</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E18:37"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="scmring2.html#E19:37"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="scmring1.html#T19" title="SCMRING1:th.19">SCMRING1:19</a></i>
<br/>.= 
 <a href="scmring1.html#K4" title="SCMRING1:func.4">SCM-Chg</a> <font color="Maroon" title="c10">w</font>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="scmring1.html#K10" title="SCMRING1:func.10">cjump_address</a> </span>)</span>,<span class="p2">(<span class="default"><a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <span class="p3">(<span class="default"><a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c10">w</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E1:37_5_2"><i><font color="Green" title="E41">A20</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
 <a href="scmring1.html#K15" title="SCMRING1:func.15">SCM-Exec-Res</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c10">w</font>
<b>by </b><i><a class="txt" href="scmring2.html#E19:37"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="scmring2.html#E20:37"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="scmring1.html#D14" title="SCMRING1:def.14">SCMRING1:def 14</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c3">i1</font></span>)</span>,<font color="Maroon" title="c9">t</font>
<b>by </b><i><a class="ref" href="scmring2.html#T12" target="_self" title="SCMRING2:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c9">t</font>
<b>by </b><i><a class="txt" href="scmring2.html#E21:37"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="ami_1.html#D8" title="AMI_1:def.8">AMI_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E3:37_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scmring2.html#E7:37"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="scmring2.html#E17:37"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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