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<b>let </b><font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</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="c6">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="c7">I</font> be   <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c4">s1</font> <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> <font color="Maroon" title="c5">s2</font> <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>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E43">A2</font></i>: 
 <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">SCMFSA9A:def 4</a><br/>

<b>assume </b><a NAME="E3:48"/><i><font color="Green" title="E44">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E4:48"/><i><font color="Green" title="E45">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</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="txt" href="scmfsa9a.html#E1:48"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="scmfsa9a.html#E2:48"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="scmfsa9a.html#T40" target="_self" title="SCMFSA9A:th.40">Th40</a></i>;<br/>
<a NAME="E5:48"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="scmfsa9a.html#E3:48"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="scmfsa6a.html#T38" title="SCMFSA6A:th.38">SCMFSA6A:38</a></i>;<br/>
<a NAME="E6:48"/><b>then </b>
( <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c4">s1</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</font> )
 
<b>by </b><i><a class="txt" href="scmfsa9a.html#E2:48"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
( <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">I</font>,<font color="Maroon" title="c5">s2</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E4:48"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="scmfsa8b.html#T8" title="SCMFSA8B:th.8">SCMFSA8B:8</a></i>;<br/>


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