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

<b>set </b><font color="Maroon" title="c1">D</font> = <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">IL</font> = the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>
<b>let </b><font color="Maroon" title="c3">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon" title="c4">a</font> be  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>  <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ;<br/><b>let </b><font color="Maroon" title="c5">s</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>





<b>given </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:25"/><i><font color="Green" title="E15">A1</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( ( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font> ) ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> $1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">IniI</font> =  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font>;<br/>
<b>set </b><font color="Maroon" title="c8">Iwhile</font> =  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">s1</font> = <font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="scmisort.html#D1" target="_self" title="SCMISORT:def.1">Def1</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green" title="E17">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) or <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:25"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E7:25"/><i><font color="Green" title="E19">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="scmfsa_9.html#S1" title="SCMFSA_9:sch.1">SCMFSA_9:sch 1</a>(<a class="txt" href="scmisort.html#E3:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="scmisort.html#E4:25"><i><font color="Green" title="E17">A3</font></i></a>);<br/></i>
<div><i><font color="Green" title="E20">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 ;<br/><a NAME="E2:25_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">i</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1"/>
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> implies  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font> );<br/>
<a NAME="E9:25"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c11">n</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3;<br/>

<b>thus </b><a NAME="E2:25_2"/>
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="ref" href="scmisort.html#T26" target="_self" title="SCMISORT:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:25"/><i><font color="Green" title="E22">A8</font></i>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa6a.html#T24" title="SCMFSA6A:th.24">SCMFSA6A:24</a></i>;<br/>
<div><i><font color="Green" title="E23">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_3"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1"/><i><font color="Green" title="E25">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ;<br/><a NAME="E2:25_3_1"/>
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_3_1"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E1:25_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:25_3_1"/><i><font color="Green" title="E27">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>set </b><font color="Maroon" title="c12">sk</font> = <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">k</font>;<br/><b>set </b><font color="Maroon" title="c13">sk1</font> = <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c14">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:25_3_1"/><i><font color="Green" title="E28">A14</font></i>: 
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E1:25_3"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="scmisort.html#E1:25_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="scmisort.html#E4:25_3_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:25_3_1"/><i><font color="Green" title="E29">A15</font></i>: 
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="scmisort.html#D1" target="_self" title="SCMISORT:def.1">Def1</a></i>;<br/><a NAME="E8:25_3_1"/><i><font color="Green" title="E30">A16</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="scmisort.html#E3:25_3_1"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/><a NAME="E9:25_3_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="scmisort.html#E3:25_3_1"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/><a NAME="E10:25_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><a NAME="E11:25_3_1"/><b>then </b><i><font color="Green" title="E31">A17</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25_3_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="scmisort.html#E8:25_3_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="scmisort.html#T19" target="_self" title="SCMISORT:th.19">Th19</a></i>;<br/><a NAME="E12:25_3_1"/><i><font color="Green" title="E32">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c15">m</font> = <font color="Maroon" title="c14">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>;<br/><b>thus </b><a NAME="E13:25_3_1"/>
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c15">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E6:25_3_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="scmisort.html#E11:25_3_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="scmisort.html#E12:25_3_1"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="scmisort.html#T27" target="_self" title="SCMISORT:th.27">Th27</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:25_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E12:25"/><i><font color="Green" title="E24">A19</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="scmisort.html#E9:25"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="scmisort.html#E11:25"><i><font color="Green" title="E23">A9</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_4_1"/>
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_1"><b>suppose </b></a><a NAME="E1:25_4_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:25_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="scmisort.html#E6:25"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/><a NAME="E3:25_4_1_1"/><b>then </b>
<font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="scmisort.html#D1" target="_self" title="SCMISORT:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:25_4_1_1"/>
(  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c5">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="ref" href="scmisort.html#T16" target="_self" title="SCMISORT:th.16">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2"><b>suppose </b></a><a NAME="E1:25_4_1_2"/><i><font color="Green" title="E25">A20</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c11">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:25_4_1_2"/><i><font color="Green" title="E26">A21</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:25_4_1_2"/><i><font color="Green" title="E27">A22</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>set </b><font color="Maroon" title="c13">si</font> = <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c12">i</font>;<br/><b>set </b><font color="Maroon" title="c14">sm</font> = <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>;<br/><b>set </b><font color="Maroon" title="c15">sm1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c16">sm2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E6:25_4_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2"><i><font color="Green" title="E26">A21</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:25_4_1_2"/><i><font color="Green" title="E28">A23</font></i>: 
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E12:25"><i><font color="Green" title="E24">A19</font></i></a></i>;<br/><a NAME="E9:25_4_1_2"/><i><font color="Green" title="E29">A24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="scmisort.html#E6:25"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/><a NAME="E10:25_4_1_2"/><i><font color="Green" title="E30">A25</font></i>: 
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="scmisort.html#D1" target="_self" title="SCMISORT:def.1">Def1</a></i>;<br/><a NAME="E11:25_4_1_2"/><i><font color="Green" title="E31">A26</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c12">i</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="scmisort.html#E5:25_4_1_2"><i><font color="Green" title="E27">A22</font></i></a></i>;<br/><a NAME="E12:25_4_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:25_4_1_2"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">i</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/><a NAME="E14:25_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><a NAME="E15:25_4_1_2"/><b>then </b><i><font color="Green" title="E32">A27</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E10:25_4_1_2"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="scmisort.html#E11:25_4_1_2"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="scmisort.html#T19" target="_self" title="SCMISORT:th.19">Th19</a></i>;<br/><a NAME="E16:25_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><a NAME="E17:25_4_1_2"/><b>then </b><i><font color="Green" title="E33">A28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="scmfsa8c.html#T18" title="SCMFSA8C:th.18">SCMFSA8C:18</a></i>;<br/><b>then </b><i><font color="Green" title="E34">A29</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmisort.html#E10:25"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmisort.html#E15:25_4_1_2"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="scmfsa6a.html#T46" title="SCMFSA6A:th.46">SCMFSA6A:46</a></i>
;<br/><a NAME="E19:25_4_1_2"/><i><font color="Green" title="E35">A30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a>  <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> </span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa8a.html#T11" title="SCMFSA8A:th.11">SCMFSA8A:11</a></i>;<br/><a NAME="E20:25_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2"><i><font color="Green" title="E28">A23</font></i></a>, <a class="ref" href="ami_1.html#T123" title="AMI_1:th.123">AMI_1:123</a></i>;<br/><a NAME="E21:25_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K26" title="AMI_1:func.26">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E17:25_4_1_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="scmfsa_9.html#T27" title="SCMFSA_9:th.27">SCMFSA_9:27</a></i>;<br/><a NAME="E22:25_4_1_2"/><b>then </b><i><font color="Green" title="E36">A31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E17:25_4_1_2"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="scmisort.html#E18:25_4_1_2"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="scmisort.html#E19:25_4_1_2"><i><font color="Green" title="E35">A30</font></i></a>, <a class="ref" href="scmfsa_9.html#T29" title="SCMFSA_9:th.29">SCMFSA_9:29</a></i>;<br/><a NAME="E23:25_4_1_2"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E9:25_4_1_2"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="scmisort.html#T16" target="_self" title="SCMISORT:th.16">Th16</a></i>;<br/><a NAME="E24:25_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 <b>by </b><i><a class="ref" href="scm_halt.html#D5" title="SCM_HALT:def.5">SCM_HALT:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E26:25_4_1_2"/><i><font color="Green" title="E37">A32</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E22:25_4_1_2"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="ami_1.html#D20" title="AMI_1:def.20">AMI_1:def 20</a></i>;<br/><a NAME="E27:25_4_1_2"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="scmisort.html#E26:25_4_1_2"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><a NAME="E28:25_4_1_2"/><b>then </b>
<font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 <b>by </b><i><a class="ref" href="ami_1.html#D20" title="AMI_1:def.20">AMI_1:def 20</a></i>;<br/><b>hence </b><a NAME="E29:25_4_1_2"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="scm_halt.html#D5" title="SCM_HALT:def.5">SCM_HALT:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c19">p</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:25_4_1_2_2"/><i><font color="Green" title="E38">A33</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E1:25_4_1_2"><i><font color="Green" title="E25">A20</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_4_1_2_2_1"/>
( <font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 or <font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:25_4_1_2_2_1_1"/><i><font color="Green" title="E39">A34</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 ;<br/><div class="add"><a NAME="E2:25_4_1_2_2_1_1"/><i><font color="Green" title="E40">A35</font></i>: 
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="scmisort.html#D1" target="_self" title="SCMISORT:def.1">Def1</a></i>;<br/><a NAME="E3:25_4_1_2_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="scmisort.html#E1:25_4_1_2_2"><i><font color="Green" title="E38">A33</font></i></a></i>;<br/><a NAME="E4:25_4_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E41">A36</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_1"><i><font color="Green" title="E40">A35</font></i></a></i>;<br/><a NAME="E5:25_4_1_2_2_1_1"/>
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c5">s</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="scmisort.html#E1:25_4_1_2_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_1"><i><font color="Green" title="E40">A35</font></i></a></i>;<br/><b>hence </b><a NAME="E6:25_4_1_2_2_1_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c20">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E1:25_4_1_2_2_1_1"><i><font color="Green" title="E39">A34</font></i></a>, <a class="txt" href="scmisort.html#E4:25_4_1_2_2_1_1"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="scmisort.html#T19" target="_self" title="SCMISORT:th.19">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:25_4_1_2_2_1_2"/><i><font color="Green" title="E39">A37</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> ) );<br/><a NAME="E2:25_4_1_2_2_1_2"/><i><font color="Green" title="E40">A38</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ;<br/><div><i><font color="Green" title="E41">A39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c21">k</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3;<br/><b>thus </b><a NAME="E1:25_4_1_2_2_1_2_1"/>
( <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c21">k</font> &amp; <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E1:25_4_1_2_2_1_2"><i><font color="Green" title="E39">A37</font></i></a>, <a class="ref" href="scmisort.html#T26" target="_self" title="SCMISORT:th.26">Th26</a></i>;<br/></div><b>end;</b></div><a NAME="E4:25_4_1_2_2_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E1:25_4_1_2_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:25_4_1_2_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:25_4_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E42">A40</font></i>: 
 ex <font color="Olive" title="b1">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>]
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2_2_1_2"><i><font color="Green" title="E41">A39</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:25_4_1_2_2_1_2"/><i><font color="Green" title="E43">A41</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">t</font>] &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">t</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_2"><i><font color="Green" title="E40">A38</font></i></a>, <a class="txt" href="scmisort.html#E6:25_4_1_2_2_1_2"><i><font color="Green" title="E42">A40</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c22">t</font> = <font color="Maroon" title="c21">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1_2_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_4_1_2_2_1_2_2_1"/>
( <font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> or <font color="Maroon" title="c22">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:25_4_1_2_2_1_2_2_1_1"/>
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c23">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:25_4_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E44">A42</font></i>: 
( <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c23">r</font> &amp; <font color="Maroon" title="c23">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2_2_1_2"><i><font color="Green" title="E43">A41</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c24">x</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:25_4_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E45">A43</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">x</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E3:25_4_1_2_2_1_2_2_1_1"><i><font color="Green" title="E44">A42</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:25_4_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E46">A44</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E22:25_4_1_2"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="scmisort.html#E3:25_4_1_2_2_1_2_2_1_1"><i><font color="Green" title="E44">A42</font></i></a>, <a class="txt" href="scmisort.html#E5:25_4_1_2_2_1_2_2_1_1"><i><font color="Green" title="E45">A43</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><a NAME="E8:25_4_1_2_2_1_2_2_1_1"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E9:25_4_1_2"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="scmisort.html#T16" target="_self" title="SCMISORT:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E9:25_4_1_2_2_1_2_2_1_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c20">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25_4_1_2_2_1_2_2_1_1"><i><font color="Green" title="E46">A44</font></i></a>, <a class="ref" href="scm_halt.html#D4" title="SCM_HALT:def.4">SCM_HALT:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_2_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:25_4_1_2_2_1_2_2_1_2"/>
<font color="Maroon" title="c22">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">m</font>
 ;<br/><div class="add"><a NAME="E2:25_4_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E44">A45</font></i>: 
<font color="Maroon" title="c22">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2_2_1_2"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">y</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E45">A46</font></i>: 
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2_2_1_2"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">y</font> = <font color="Maroon" title="c23">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:25_4_1_2_2_1_2_2_1_2"/>
<font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E4:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E45">A46</font></i></a></i>;<br/><a NAME="E7:25_4_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E46">A47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c25">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E47">A48</font></i>: 
( <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c25">z</font> &amp; <font color="Maroon" title="c25">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2_2_1_2"><i><font color="Green" title="E43">A41</font></i></a></i>;<br/><a NAME="E10:25_4_1_2_2_1_2_2_1_2"/>
<font color="Maroon" title="c24">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">t</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E4:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E45">A46</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:25_4_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E48">A49</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2_2_1_2"><i><font color="Green" title="E43">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c26">Dy</font> = <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c24">y</font>;<br/><b>set </b><font color="Maroon" title="c27">Dt</font> = <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font>;<br/><a NAME="E12:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E49">A50</font></i>: 
<span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c24">y</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c24">y</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E4:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E45">A46</font></i></a>, <a class="ref" href="scmisort.html#D1" target="_self" title="SCMISORT:def.1">Def1</a></i>;<br/><a NAME="E13:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E50">A51</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c24">y</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c24">y</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="scmisort.html#E11:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E48">A49</font></i></a></i>;<br/><a NAME="E14:25_4_1_2_2_1_2_2_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c24">y</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="scmisort.html#E11:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E48">A49</font></i></a></i>;<br/><a NAME="E15:25_4_1_2_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c24">y</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><a NAME="E16:25_4_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E51">A52</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E12:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E49">A50</font></i></a>, <a class="txt" href="scmisort.html#E13:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E50">A51</font></i></a>, <a class="ref" href="scmisort.html#T19" target="_self" title="SCMISORT:th.19">Th19</a></i>;<br/><b>set </b><font color="Maroon" title="c28">z2</font> = <font color="Maroon" title="c25">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>;<br/><div><i><font color="Green" title="E52">A53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1_2_2_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_4_1_2_2_1_2_2_1_2_1"/><i><font color="Green" title="E53">A54</font></i>: 
<font color="Maroon" title="c25">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font>
 ;<br/><div><i><font color="Green" title="E54">A55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1_2_2_1_2_2_1_2_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c29">k</font> = <font color="Maroon" title="c25">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>;<br/><b>thus </b><a NAME="E1:25_4_1_2_2_1_2_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">q</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E46">A47</font></i></a>, <a class="txt" href="scmisort.html#E9:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E47">A48</font></i></a>, <a class="txt" href="scmisort.html#E16:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E51">A52</font></i></a>, <a class="txt" href="scmisort.html#E1:25_4_1_2_2_1_2_2_1_2_1"><i><font color="Green" title="E53">A54</font></i></a>, <a class="ref" href="scmisort.html#T27" target="_self" title="SCMISORT:th.27">Th27</a></i>;<br/></div><b>end;</b></div><a NAME="E3:25_4_1_2_2_1_2_2_1_2_1"/>
<font color="Maroon" title="c22">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E44">A45</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:25_4_1_2_2_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c22">t</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">t</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25_4_1_2_2_1_2"><i><font color="Green" title="E43">A41</font></i></a>, <a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_2_2_1_2_1"><i><font color="Green" title="E54">A55</font></i></a></i>;<br/><b>hence </b><a NAME="E5:25_4_1_2_2_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c29">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E53">A56</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">w</font>
 <b>by </b><i><a class="txt" href="scmisort.html#E9:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E47">A48</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">w</font> = <font color="Maroon" title="c29">w</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E21:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E54">A57</font></i>: 
<font color="Maroon" title="c30">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="ami_1.html#K29" title="AMI_1:func.29">LifeSpan</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <b>by </b><i><a class="txt" href="scmisort.html#E17:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E52">A53</font></i></a>, <a class="txt" href="scmisort.html#E19:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E55">A58</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c20">q</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c30">w</font>
<b>by </b><i><a class="txt" href="scmisort.html#E9:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E47">A48</font></i></a>, <a class="txt" href="scmisort.html#E19:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c30">w</font>
<b>by </b><i><a class="txt" href="scmisort.html#E7:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E46">A47</font></i></a>, <a class="txt" href="scmisort.html#E9:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E47">A48</font></i></a>, <a class="txt" href="scmisort.html#E16:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E51">A52</font></i></a>, <a class="ref" href="scmisort.html#T27" target="_self" title="SCMISORT:th.27">Th27</a></i>
;<br/><a NAME="E23:25_4_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E56">A59</font></i>: 
( <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <span class="p1">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font> )
 <b>by </b><i><a class="txt" href="scmisort.html#E8:25"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E44">A45</font></i></a></i>;<br/><a NAME="E24:25_4_1_2_2_1_2_2_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c22">t</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E7:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="scmisort.html#E2:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E44">A45</font></i></a></i>;<br/><a NAME="E25:25_4_1_2_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmisort.html#K1" title="SCMISORT:func.1">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">I</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scmisort.html#E2:25"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E26:25_4_1_2_2_1_2_2_1_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c20">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmisort.html#E21:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E54">A57</font></i></a>, <a class="txt" href="scmisort.html#E22:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E55">A58</font></i></a>, <a class="txt" href="scmisort.html#E23:25_4_1_2_2_1_2_2_1_2"><i><font color="Green" title="E56">A59</font></i></a>, <a class="ref" href="scmisort.html#T19" target="_self" title="SCMISORT:th.19">Th19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:25_4_1_2_2_1_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c20">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E31:25_4_1_2"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="scm_halt.html#D4" title="SCM_HALT:def.4">SCM_HALT:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:25"/>
(  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R2" title="SCM_HALT:pred.2">is_halting_onInit</a> <font color="Maroon" title="c5">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scm_halt.html#R1" title="SCM_HALT:pred.1">is_closed_onInit</a> <font color="Maroon" title="c5">s</font> )
 ;<br/>


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