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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">I</font> be  <a href="scmfsa6b.html#V2">parahalting</a> <a href="scmfsa6a.html#NM1">Macro-Instruction</a>;<br/>



<b>assume </b><a NAME="E1:111"/><i><font color="Green">E31</font></i>: 
 <a href="scmfsa6a.html#K3">Initialized</a> <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:111"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">s</font> is <a href="ami_1.html#V9">halting</a>
 
<b>by </b><i><a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/>
<a NAME="E3:111"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">I</font> <a href="ami_1.html#K17">+*</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12">Start-At</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="scmfsa6b.html#T8">SCMFSA6B:8</a></i>;<br/>
<b>set </b><font color="Maroon">s2</font> = <font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">A</font> = the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>
<b>set </b><font color="Maroon">l1</font> =  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">s</font></span>)</span></span>)</span>;<br/>
<div><b>hereby </b>
<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> ;<br/><b>assume </b><a NAME="E1:111_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_1.html#NK3" target="_self">LifeSpan</a> <font color="Maroon">s</font>
 ;<br/><b>set </b><font color="Maroon">lk</font> =  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span>;<br/><a NAME="E2:111_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p3">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font> <a href="funct_7.html#R1">equal_outside</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i>, <a class="ref" href="scmfsa8c.html#T39" target="_self">Th39</a>, <a class="ref" href="scmfsa8c.html#T41" target="_self">Th41</a>, <a class="ref" href="scmfsa8c.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E3:111_1"/><i><font color="Green">E45</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="scmfsa8c.html#T39" target="_self">Th39</a>, <a class="ref" href="scmfsa6b.html#D2">SCMFSA6B:def 2</a></i>;<br/><a NAME="E4:111_1"/><i><font color="Green">E46</font></i>: 
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa8c.html#T42" target="_self">Th42</a>, <a class="ref" href="scmfsa6a.html#T29">SCMFSA6A:29</a></i>;<br/><a NAME="E5:111_1"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E6:111_1"/><i><font color="Green">E49</font></i>: 
 <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 ;<br/><i><font color="Green">E50</font></i>:  <a href="ami_1.html#K8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p4">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ami_1.html#D17">AMI_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="ami_5.html#K8">+*</a> <span class="p2">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span></span>)</span> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T54">AMI_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/><a NAME="E8:111_1"/>
<span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6">IC</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" target="_self">loop</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:111_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>


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