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

<b>let </b><font color="Maroon" title="c2">s</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> <br/> for <font color="Olive" title="b2">a</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>   st <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">does_not_destroy</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font> holds <br/>for <font color="Olive" title="b3">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  <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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Olive" title="b3">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b2">a</font></span><br/><b>let </b><font color="Maroon" title="c3">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"> for <font color="Olive" title="b1">a</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>   st <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">does_not_destroy</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font> holds <br/>for <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>  holds  <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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Olive" title="b1">a</font></span><br/><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">does_not_destroy</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</font> implies 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  <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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> )</span><br/>





<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">does_not_destroy</a> <font color="Maroon" title="c4">a</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="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</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  <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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<b>assume </b><a NAME="E2:46"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c2">s</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  <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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E3:46"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></i>;<br/>
<a NAME="E4:46"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">I</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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#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="sf_mastr.html#T64" title="SF_MASTR:th.64">SF_MASTR:64</a></i>;<br/>
<a NAME="E5:46"/><b>then </b>
<font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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="funct_4.html#T33" title="FUNCT_4:th.33">FUNCT_4:33</a></i>;<br/>
<a NAME="E6:46"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="scmfsa7b.html#E3:46"><i><font color="Green" title="E27">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="E7:46"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c4">a</font> <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="c2">s</font> &amp; not <font color="Maroon" title="c4">a</font> <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"><font color="Maroon" title="c3">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="scmfsa6b.html#T12" title="SCMFSA6B:th.12">SCMFSA6B:12</a>, <a class="ref" href="scmfsa_2.html#T66" title="SCMFSA_2:th.66">SCMFSA_2:66</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,$1</span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>;<br/>
<a NAME="E8:46"/><i><font color="Green" title="E30">A6</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="46_1"><b>proof </b></a><div class="add">

<b>thus </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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E7:46"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</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>
<div><i><font color="Green" title="E31">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>set </b><font color="Maroon" title="c6">l</font> =  <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"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Maroon" title="c5">k</font></span>)</span>;<br/><a NAME="E2:46_2"/><i><font color="Green" title="E33">A9</font></i>: 
 <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"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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>,<font color="Maroon" title="c5">k</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="c3">I</font>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E2:46"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="scmfsa7b.html#D7" target="_self" title="SCMFSA7B:def.7">Def7</a></i>;<br/><a NAME="E3:46_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E6:46"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E4:46_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E2:46_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:46_2"/><b>then </b><i><font color="Green" title="E34">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p3">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">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="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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="scmfsa7b.html#R3" title="SCMFSA7B:pred.3">does_not_destroy</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:46"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="scmfsa7b.html#D4" target="_self" title="SCMFSA7B:def.4">Def4</a></i>;<br/><b>thus </b><a NAME="E6:46_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1"><b>proof </b></a><div class="add">

<b>thus </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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p4">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p5">(<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="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="ami_1.html#T14" title="AMI_1:th.14">AMI_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p0">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</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></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span>,<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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="ami_1.html#D18" title="AMI_1:def.18">AMI_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p0">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</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></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">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></span>)</span></span>)</span>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>,<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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="ami_1.html#D17" title="AMI_1:def.17">AMI_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p3">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">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></span>)</span></span>)</span>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>,<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="c2">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="ami_1.html#K18" title="AMI_1:func.18">+*</a> <span class="p5">(<span class="default"><a href="ami_1.html#K28" title="AMI_1:func.28">Start-At</a> <span class="p0">(<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>,<font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="ami_1.html#T54" title="AMI_1:th.54">AMI_1:54</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="scmfsa_2.html#K19" title="SCMFSA_2:func.19">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E1:46_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="scmfsa7b.html#E5:46_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="scmfsa7b.html#T26" target="_self" title="SCMFSA7B:th.26">Th26</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></div><b>end;</b></div>
<b>thus </b><a NAME="E10:46"/>
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  <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="scmfsa7b.html#E8:46"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="scmfsa7b.html#E9:46"><i><font color="Green" title="E31">A7</font></i></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>
