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<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>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">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>





<b>assume </b><a NAME="E1:54"/><i><font color="Green">E32</font></i>: 
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="card_3.html#K4">product</a> the <a href="ami_1.html#U5">Object-Kind</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> , <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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="Olive">k</font></span>)</span> or <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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="Olive">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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="Olive">k</font></span>)</span> <a href="scmfsa_2.html#K20">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" target="_self">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="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/>

<a NAME="E2:54"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <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" target="_self">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="Olive">k</font>
 
<b>by </b><i><a class="ref" href="scmfsa7b.html#T24">SCMFSA7B:24</a></i>;<br/>
<a NAME="E3:54"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <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" target="_self">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="Olive">k</font>
 
<b>by </b><i><a class="ref" href="scmfsa7b.html#T25">SCMFSA7B:25</a></i>;<br/>
<b>hence </b><a NAME="E4:54"/>
(  <a href="scmfsa_9.html#K2" target="_self">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" target="_self">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="ref" href="scmfsa_9.html#T10" target="_self">Th10</a></i>;<br/>


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