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<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="sf_mastr.html#V1">read-write</a>  <a href="scmfsa_2.html#M1">Int-Location</a> ;<br/><b>let </b><font color="Maroon">I</font> be  <a href="scmfsa6b.html#V2">parahalting</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green">E33</font></i>: 
 <a href="scmfsa9a.html#R4" target="_self">WithVariantWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:38"/>
 <a href="scmfsa9a.html#R3" target="_self">ProperBodyWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="scmfsa9a.html#D4" target="_self">SCMFSA9A:def 4</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:38_1"/>
( <font color="Maroon">I</font> <a href="scmfsa7b.html#R5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">I</font> <a href="scmfsa7b.html#R6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5">StepWhile&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font>,<font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="scmfsa7b.html#T24">SCMFSA7B:24</a>, <a class="ref" href="scmfsa7b.html#T25">SCMFSA7B:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
(  <a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font> <a href="scmfsa7b.html#R6">is_halting_on</a> <font color="Maroon">s</font> &amp;  <a href="scmfsa_9.html#K2">while&gt;0</a> <font color="Maroon">a</font>,<font color="Maroon">I</font> <a href="scmfsa7b.html#R5">is_closed_on</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="scmfsa9a.html#E1:38"><i><font color="Green">E33</font></i></a>, </i>;<br/>


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