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

<b>let </b><font color="Maroon">s1</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="sf_mastr.html#V1">read-write</a>  <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">I</font> be   <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E33</font></i>: 
 <a href="scmfsa9a.html#R1" target="_self">ProperBodyWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">s1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">ST1</font> =  <a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font>;<br/>
<b>set </b><font color="Maroon">ST2</font> =  <a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font>;<br/>
<b>set </b><font color="Maroon">WH</font> =  <a href="scmfsa_9.html#K1">while=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> = 
<font color="Maroon">s1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_9.html#D4">SCMFSA_9:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> 0</span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="scmfsa_9.html#D4">SCMFSA_9:def 4</a></i>
;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E5:26"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
 ;<br/>

<a NAME="E2:26_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="scmfsa6a.html#T38">SCMFSA6A:38</a></i>;<br/>
<b>set </b><font color="Maroon">ST1kI</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">ST2kI</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i>, , , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_2"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:26_2_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">I</font> <a href="scmfsa7b.html#R5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">I</font> <a href="scmfsa7b.html#R6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><i><font color="Green">E83</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K1">while=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_9.html#D4">SCMFSA_9:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E4:26_2_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">I</font> <a href="scmfsa7b.html#R5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">I</font> <a href="scmfsa7b.html#R6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="ref" href="scmfsa8b.html#T8">SCMFSA8B:8</a></i>;<br/><i><font color="Green">E85</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K1">while=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_9.html#D4">SCMFSA_9:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i>, , <a class="txt" href="scmfsa9a.html#E1:26"><i><font color="Green">E33</font></i></a>, </i>
;<br/><i><font color="Green">E86</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i><a class="ref" href="scmfsa8a.html#T11">SCMFSA8A:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="scmfsa8a.html#T11">SCMFSA8A:11</a></i>
;<br/><a NAME="E7:26_2_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><a NAME="E8:26_2_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><a NAME="E9:26_2_1_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa8a.html#T11">SCMFSA8A:11</a></i>;<br/><a NAME="E10:26_2_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">I</font> <a href="scmfsa7b.html#R5">is_closed_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="scmfsa8b.html#T6">SCMFSA8B:6</a></i>;<br/><a NAME="E11:26_2_1_2"/>
<font color="Maroon">I</font> <a href="scmfsa7b.html#R6">is_halting_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scmfsa8b.html#T8">SCMFSA8B:8</a></i>;<br/><a NAME="E12:26_2_1_2"/><b>then </b>
 <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="scmfsa9a.html#E4:26"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="scmfsa8c.html#T44">SCMFSA8C:44</a></i>;<br/><b>hence </b><a NAME="E13:26_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4">StepWhile=0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s2</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="finsub_1.html#K1">\/</a> <a href="scmfsa_2.html#K3">FinSeq-Locations</a> </span>)</span>
 <b>by </b><i>, <a class="txt" href="scmfsa9a.html#E2:26"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="scmfsa9a.html#E4:26"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="scmfsa8c.html#T43">SCMFSA8C:43</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E6:26"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
