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

<b>set </b><font color="Maroon">SW</font> =  <a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">S</font> = <font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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>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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:45_1_1"/><i><font color="Green">E36</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>   holds <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">k</font></span>)</span> or <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E3:45_1_1"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E3:45_1_1"/><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>   holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1) <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) or <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] )
 
<b>by </b><i><a class="txt" href="scmfsa9a.html#E5:45_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 <b>and </b><br/><a NAME="E6:45_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="scmfsa9a.html#S1" target="_self">SCMFSA9A:sch 1</a>(<a class="txt" href="scmfsa9a.html#E6:45_1_1"><i><font color="Green">E42</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E7:45_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>

<b>thus </b><a NAME="E8:45_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E9:45_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Olive">n</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T6" target="_self">Th6</a></i>;<br/>

<a NAME="E10:45_1_1"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> <font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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="E11:45_1_1"/><i><font color="Green">E81</font></i>: 
 <a href="scmfsa9a.html#R3" target="_self">ProperBodyWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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> <font color="Olive">k</font> );<br/>
<a NAME="E12:45_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>take </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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;<br/>

<b>thus </b><a NAME="E2:45_1_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="scmfsa_9.html#T51">SCMFSA_9:51</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:45_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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="sf_mastr.html#T65">SF_MASTR:65</a></i>;<br/>
<div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>now </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="E1:45_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_1_2_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:45_1_1_2_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:45_1_1_2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:45_1_1_2_1"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>set </b><font color="Maroon">sk</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">sk1</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:45_1_1_2_1"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="scmfsa9a.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:45_1_1_2_1"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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="p1">(<span class="default"><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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</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="nat_1.html#K1">+</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa_9.html#D5">SCMFSA_9:def 5</a></i>;<br/><a NAME="E8:45_1_1_2_1"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T6" target="_self">Th6</a>, <a class="ref" href="scmfsa9a.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E9:45_1_1_2_1"/><b>then </b>
( <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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E10:45_1_1_2_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</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="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T11" target="_self">Th11</a>, <a class="ref" href="scmfsa9a.html#T12" target="_self">Th12</a>, <a class="ref" href="scmfsa_9.html#T47">SCMFSA_9:47</a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="nat_1.html#K1">+</a> 3</span>)</span>;<br/><b>thus </b><a NAME="E11:45_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T10" target="_self">Th10</a>, , <a class="ref" href="scmfsa_9.html#T52">SCMFSA_9:52</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:45_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E15:45_1_1"/><i><font color="Green">E112</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="scmfsa9a.html#T7" target="_self">Th7</a>);<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_3"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3_1"><b>suppose </b></a><a NAME="E1:45_1_1_3_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><i><font color="Green">E114</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> = 
<font color="Maroon">s</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="scmfsa8a.html#T11">SCMFSA8A:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</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="scmfsa_9.html#D5">SCMFSA_9:def 5</a></i>
;<br/><a NAME="E3:45_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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="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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</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>hence </b><a NAME="E4:45_1_1_3_1"/>
<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"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;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"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;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></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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</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="scmfsa9a.html#T8" target="_self">Th8</a>, <a class="txt" href="scmfsa9a.html#E1:45"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3_2"><b>suppose </b></a><a NAME="E1:45_1_1_3_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:45_1_1_3_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">si</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">sm</font> = <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font>;<br/><b>set </b><font color="Maroon">sm1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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/><a NAME="E5:45_1_1_3_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:45_1_1_3_2"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E8:45_1_1_3_2"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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="p1">(<span class="default"><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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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="nat_1.html#K1">+</a> 3</span>)</span>
 <b>by </b><i>, <a class="ref" href="scmfsa_9.html#D5">SCMFSA_9:def 5</a></i>;<br/><a NAME="E9:45_1_1_3_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:45_1_1_3_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E11:45_1_1_3_2"/><b>then </b>
( <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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E12:45_1_1_3_2"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> 0
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T9" target="_self">Th9</a>, <a class="ref" href="scmfsa9a.html#T10" target="_self">Th10</a>, <a class="ref" href="scmfsa_9.html#T47">SCMFSA_9:47</a></i>;<br/><i><font color="Green">E121</font></i>:  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15">AMI_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa9a.html#T11" target="_self">Th11</a>, <a class="ref" href="sf_mastr.html#T66">SF_MASTR:66</a></i>
;<br/><a NAME="E14:45_1_1_3_2"/><i><font color="Green">E122</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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</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/><a NAME="E15:45_1_1_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T8" target="_self">Th8</a>, <a class="ref" href="scmfsa6b.html#T17">SCMFSA6B:17</a></i>;<br/><a NAME="E16:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="scmfsa_9.html#T27">SCMFSA_9:27</a></i>;<br/><a NAME="E17:45_1_1_3_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T12" target="_self">Th12</a>, , <a class="ref" href="scmfsa_9.html#T29">SCMFSA_9:29</a></i>;<br/><a NAME="E18:45_1_1_3_2"/>
 <a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="scmfsa_9.html#T43">SCMFSA_9:43</a></i>;<br/><a NAME="E19:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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> is <a href="ami_1.html#V9">halting</a>
 <b>by </b><i><a class="ref" href="scmfsa7b.html#D8">SCMFSA7B:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:45_1_1_3_2"/><i><font color="Green">E124</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <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"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i>, <a class="ref" href="ami_1.html#D20">AMI_1:def 20</a></i>;<br/><a NAME="E22:45_1_1_3_2"/><i><font color="Green">E125</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <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"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="scmfsa9a.html#T8" target="_self">Th8</a>, , <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><a NAME="E23:45_1_1_3_2"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <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"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/><span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font></span>)</span> <a href="ami_1.html#K17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p5">(<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="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="scmfsa9a.html#T8" target="_self">Th8</a>, <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E25:45_1_1_3_2"/>
<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"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;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"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa_9.html#K2">while&gt;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></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#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">m</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="txt" href="scmfsa9a.html#E1:45"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
