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

<b>let </b><font color="Maroon" title="c3">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>let </b><font color="Maroon" title="c4">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/>



<b>set </b><font color="Maroon" title="c5">s1</font> = <font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">s2</font> = <font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p2">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">m1</font> =  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c4">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s</font> )
 ;<br/>

<a NAME="E2:45"/>
 <a href="ami_5.html#K6" title="AMI_5:func.6">ProgramPart</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font>
 
<b>by </b><i><a class="ref" href="ami_5.html#T72" title="AMI_5:th.72">AMI_5:72</a></i>;<br/>
<a NAME="E3:45"/><b>then </b><i><font color="Green" title="E40">A2</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ami_5.html#K6" title="AMI_5:func.6">ProgramPart</a> <span class="p2">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="scmfsa8a.html#T34" target="_self" title="SCMFSA8A:th.34">Th34</a>, <a class="ref" href="scmfsa6a.html#T14" title="SCMFSA6A:th.14">SCMFSA6A:14</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green" title="E41">A3</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="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p0">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="ami_5.html#K6" title="AMI_5:func.6">ProgramPart</a> <span class="p3">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa8a.html#E1:45"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="scmfsa8a.html#T36" target="_self" title="SCMFSA8A:th.36">Th36</a></i>;<br/>
<div><i><font color="Green" title="E42">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E2:45_1"/><b>then </b>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p3">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 <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="E3:45_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">n</font>,<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">n</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</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="scmfsa8a.html#E1:45"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="scmfsa8a.html#T35" target="_self" title="SCMFSA8A:th.35">Th35</a></i>;<br/><a NAME="E4:45_1"/><b>then </b>
 <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#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa6a.html#T29" title="SCMFSA6A:th.29">SCMFSA6A:29</a></i>;<br/><b>hence </b><a NAME="E5:45_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#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c8">n</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="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E1:45"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="scmfsa8a.html#E3:45"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="scmfsa7b.html#D7" title="SCMFSA7B:def.7">SCMFSA7B:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:45"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st not  <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#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p5">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p0">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Olive" title="b1">n</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="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<b>thus </b><a NAME="E7:45"/>
 <a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font> <a href="scmfsa8a.html#R1" title="SCMFSA8A:pred.1">is_pseudo-closed_on</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E4:45"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="scmfsa8a.html#E5:45"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="scmfsa8a.html#D3" target="_self" title="SCMFSA8A:def.3">Def3</a></i>;<br/>

<b>hence </b><a NAME="E8:45"/>
 <a href="scmfsa8a.html#K3" title="SCMFSA8A:func.3">pseudo-LifeSpan</a> <font color="Maroon" title="c3">s</font>,<span class="p1">(<span class="default"><a href="scmfsa6a.html#K1" title="SCMFSA6A:func.1">Directed</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_1.html#NK3" title="SCM_1:NK.3">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_5.html#K8" title="AMI_5:func.8">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="ami_1.html#K17" title="AMI_1:func.17">+*</a> <span class="p4">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">Start-At</a> <span class="p5">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="scmfsa8a.html#E4:45"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="scmfsa8a.html#E6:45"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="scmfsa8a.html#D5" target="_self" title="SCMFSA8A:def.5">Def5</a></i>;<br/>


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