<?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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="scmfsa_2.html#K3" title="SCMFSA_2:func.3">FinSeq-Locations</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">A</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b2">k</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  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><b>let </b><font color="Maroon" title="c5">I</font> be   <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> &amp;  ex <font color="Olive" title="b1">k</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  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>



<b>assume </b><a NAME="E1:90"/><i><font color="Green" title="E74">A1</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s1</font> or  not <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> or  not <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> or for <font color="Olive" title="b1">k</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>   holds not  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E2:90"/><i><font color="Green" title="E75">A2</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font> or  not <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> or for <font color="Olive" title="b1">k</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>   holds not  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E3:90"/><i><font color="Green" title="E76">A3</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font> or for <font color="Olive" title="b1">k</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>   holds not  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E4:90"/><i><font color="Green" title="E77">A4</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( for <font color="Olive" title="b1">k</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>   holds not  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or  <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>given </b><font color="Maroon" title="c6">k</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> <b> such that </b><a NAME="E6:90"/><i><font color="Green" title="E78">A5</font></i>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<b>set </b><font color="Maroon" title="c7">s3</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>;<br/>
<a NAME="E7:90"/><i><font color="Green" title="E79">A6</font></i>: 
<font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="#E4:90"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>;<br/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span> = 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c4">s2</font>
<b>by </b><i><a class="txt" href="#E6:90"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="scmfsa8a.html#T6" title="SCMFSA8A:th.6">SCMFSA8A:6</a></i>
<br/>.= 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s2</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="#E7:90"><i><font color="Green" title="E79">A6</font></i></a>, <a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>
<br/>.= 
 <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="ref" href="ami_1.html#T111" title="AMI_1:th.111">AMI_1:111</a></i>
;<br/>
<a NAME="E9:90"/><b>then </b>
(  <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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </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="ref" href="ami_1.html#T94" title="AMI_1:th.94">AMI_1:94</a></i>;<br/>
<a NAME="E10:90"/><b>then </b><i><font color="Green" title="E80">A7</font></i>: 
<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> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T88" title="FUNCOP_1:th.88">FUNCOP_1:88</a></i>;<br/>
<a NAME="E11:90"/><i><font color="Green" title="E81">A8</font></i>: 
 <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K15" title="SCMNORM:func.15">DataPart</a> <font color="Maroon" title="c4">s2</font>
 
<b>by </b><i><a class="txt" href="#E6:90"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="scmfsa8a.html#T6" title="SCMFSA8A:th.6">SCMFSA8A:6</a></i>;<br/>
<a NAME="E12:90"/>
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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>
 
<b>by </b><i><a class="ref" href="scmfsa8a.html#T9" title="SCMFSA8A:th.9">SCMFSA8A:9</a></i>;<br/>
<a NAME="E13:90"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="txt" href="#E3:90"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:90"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="ami_1.html#T86" title="AMI_1:th.86">AMI_1:86</a></i>;<br/>
<a NAME="E15:90"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p1">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p2">(<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="#E10:90"><i><font color="Green" title="E80">A7</font></i></a>, <a class="ref" href="funct_4.html#T92" title="FUNCT_4:th.92">FUNCT_4:92</a></i>;<br/>
<a NAME="E16:90"/><b>then </b><i><font color="Green" title="E82">A9</font></i>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>;<br/>
<a NAME="E17:90"/><i><font color="Green" title="E83">A10</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p2">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="#E3:90"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span>,<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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">I</font></span><br/><a NAME="E1:90_2"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><b>hence </b><a NAME="E2:90_2"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <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"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span>,<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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">I</font>
 <b>by </b><i><a class="txt" href="#E1:90"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="#E17:90"><i><font color="Green" title="E83">A10</font></i></a>, <a class="ref" href="scmfsa7b.html#D7" title="SCMFSA7B:def.7">SCMFSA7B:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E19:90"/><b>then </b><i><font color="Green" title="E84">A11</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="#E16:90"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="scmfsa7b.html#D7" title="SCMFSA7B:def.7">SCMFSA7B:def 7</a></i>;<br/>
<a NAME="E20:90"/><i><font color="Green" title="E85">A12</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="#E2:90"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="#E17:90"><i><font color="Green" title="E83">A10</font></i></a>, <a class="ref" href="scmfsa7b.html#D8" title="SCMFSA7B:def.8">SCMFSA7B:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">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> <b> such that </b><br/><a NAME="E22:90"/><i><font color="Green" title="E86">A13</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c8">n</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="ref" href="ami_1.html#D20" title="AMI_1:def.20">AMI_1:def 20</a></i>;<br/>
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <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"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c8">n</font></span>)</span> = 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>
<br/>.= 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c8">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E22:90"><i><font color="Green" title="E86">A13</font></i></a>, <a class="ref" href="ami_1.html#T52" title="AMI_1:th.52">AMI_1:52</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
;<br/>
<a NAME="E24:90"/><b>then </b>
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="#E22:90"><i><font color="Green" title="E86">A13</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="E25:90"/><b>then </b>
<font color="Maroon" title="c5">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="#E16:90"><i><font color="Green" title="E82">A9</font></i></a>, <a class="ref" href="scmfsa7b.html#D8" title="SCMFSA7B:def.8">SCMFSA7B:def 8</a></i>;<br/>
<a NAME="E26:90"/><b>then </b><i><font color="Green" title="E87">A14</font></i>: 
 <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c6">k</font></span>)</span>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="#E7:90"><i><font color="Green" title="E79">A6</font></i></a>, <a class="txt" href="#E11:90"><i><font color="Green" title="E81">A8</font></i></a>, <a class="txt" href="#E16:90"><i><font color="Green" title="E82">A9</font></i></a>, <a class="txt" href="#E19:90"><i><font color="Green" title="E84">A11</font></i></a>, <a class="ref" href="scmfsa8c.html#T101" target="_self" title="SCMFSA8C:th.101">Th101</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">k</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> <b> such that </b><br/><a NAME="E28:90"/><i><font color="Green" title="E88">A15</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c9">k</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="#E20:90"><i><font color="Green" title="E85">A12</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="E29:90"/>
<font color="Maroon" title="c3">s1</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c9">k</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="#E28:90"><i><font color="Green" title="E88">A15</font></i></a>, <a class="ref" href="ami_1.html#T54" title="AMI_1:th.54">AMI_1:54</a></i>;<br/>
<b>hence </b><a NAME="E30:90"/>
 <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c3">s1</font>, <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <font color="Maroon" title="c4">s2</font> <a href="funct_7.html#R1" title="FUNCT_7:pred.1">equal_outside</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="#E26:90"><i><font color="Green" title="E87">A14</font></i></a>, <a class="ref" href="ami_1.html#T57" title="AMI_1:th.57">AMI_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
