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<b>let </b><font color="Maroon" title="c3">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="c4">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>given </b><font color="Maroon" title="c5">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#U5" title="AMI_1:sel.5">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> <b> such that </b><a NAME="E1:55"/><i><font color="Green" title="E47">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being  <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>  holds <br/> ( ( <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> or <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Olive" title="b1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 ) &amp; ( <font color="Olive" title="b1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">t</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:55_1_1"/>
( ( <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> or <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <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="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 ) &amp; ( <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="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="scmfsa_9.html#E1:55"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:55_1_1"/>
( ( <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> or <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <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="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 ) &amp; ( <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="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <font color="Maroon" title="c5">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#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 <b>by </b><i><a class="ref" href="scmfsa_9.html#T50" target="_self" title="SCMFSA_9:th.50">Th50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:55_1"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c6">t</font>
 <b>by </b><i><a class="ref" href="scmfsa_9.html#T54" target="_self" title="SCMFSA_9:th.54">Th54</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
 <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> is <a href="scmfsa6b.html#V2" title="SCMFSA6B:attr.2">parahalting</a>
 <b>by </b><i><a class="ref" href="scmfsa7b.html#T25" title="SCMFSA7B:th.25">SCMFSA7B:25</a></i>;<br/>


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