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<b>let </b><font color="Maroon" title="c2">I</font> be  <a href="scmfsa6b.html#V2" title="SCMFSA6B:attr.2">parahalting</a> <a href="scmfsa6a.html#NM1" title="SCMFSA6A:NM.1">Macro-Instruction</a>;<br/><b>let </b><font color="Maroon" title="c3">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="c4">s</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>assume </b><a NAME="E1:32"/><i><font color="Green" title="E26">A1</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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 <br/> ( ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b2">k</font></span>)</span> or <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ;<br/>

<a NAME="E2:32"/><i><font color="Green" title="E27">A2</font></i>: 
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" title="c2">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#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font>
 
<b>by </b><i><a class="ref" href="scmfsa7b.html#T24" title="SCMFSA7B:th.24">SCMFSA7B:24</a></i>;<br/>
<a NAME="E3:32"/>
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" title="c2">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#K4" title="SCMFSA_9:func.4">StepWhile=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="ami_1.html#K10" title="AMI_1:func.10">.</a> <font color="Olive" title="b1">k</font>
 
<b>by </b><i><a class="ref" href="scmfsa7b.html#T25" title="SCMFSA7B:th.25">SCMFSA7B:25</a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
(  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c4">s</font> &amp;  <a href="scmfsa_9.html#K1" title="SCMFSA_9:func.1">while=0</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c4">s</font> )
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E1:32"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="scmfsa_9.html#E2:32"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="scmfsa_9.html#T32" target="_self" title="SCMFSA_9:th.32">Th32</a></i>;<br/>


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