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<div class="add">

<b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>assume </b><a NAME="E1:71"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">R</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be  <a href="ami_1.html#V11">autonomic</a>  <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E2:71"/><i><font color="Green">E57</font></i>: 
 <a href="ami_1.html#K2">IC</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> is <a href="ami_1.html#V12">halting</a>
 ;<br/><b>thus </b><a NAME="E2:71_1"/>
 <a href="scmring4.html#K3" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> is <a href="ami_1.html#V12">halting</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D26">AMI_1:def 26</a><br/>

<b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="scmring4.html#K3" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 ;<br/>

<b>set </b><font color="Maroon">s</font> = <font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <font color="Maroon">p</font>;<br/>
<a NAME="E2:71_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/>
<a NAME="E3:71_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <font color="Maroon">p</font> is <a href="ami_1.html#V9">halting</a>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:71"><i><font color="Green">E57</font></i></a>, <a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:71_1_1"/><i><font color="Green">E74</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">t</font> <a href="ami_5.html#K8">+*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#D20">AMI_1:def 20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s3</font> = <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span> as   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="ami_5.html#T82">AMI_5:82</a></i>;<br/>
<a NAME="E7:71_1_1"/>
<font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font>
 
;<br/>
<b>then </b><i><font color="Green">E76</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> <font color="Maroon">t</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">u</font></span>)</span> = 
 <a href="amistd_2.html#K8">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K5">halt</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i>, , <a class="txt" href="scmring4.html#E12"><i><font color="Green">Lemma58</font></i></a>, <a class="txt" href="scmring4.html#E13"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="scmring4.html#T8" target="_self">Th8</a>, <a class="txt" href="scmring4.html#E11"><i><font color="Green">Lemma53</font></i></a></i>
<br/>.= 
 <a href="ami_1.html#K5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="amistd_2.html#T29">AMISTD_2:29</a></i>
;<br/>
<b>take </b>
<font color="Maroon">u</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D20">AMI_1:def 20</a><br/>

<b>thus </b><a NAME="E9:71_1_1"/>
 <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> <font color="Maroon">t</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring4.html#E1:71_1"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:71"/><i><font color="Green">E77</font></i>: 
 <a href="scmring4.html#K3" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> is <a href="ami_1.html#V12">halting</a>
 ;<br/>

<b>let </b><font color="Maroon">t</font> be   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D26">AMI_1:def 26</a><br/>

<b>assume </b><a NAME="E5:71"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmring4.html#K3" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span> as   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> ;<br/>
<a NAME="E7:71"/><i><font color="Green">E79</font></i>: 
 <a href="scmring4.html#K3" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> <a href="ami_5.html#K8">+*</a> <span class="p1">(<span class="default"><a href="scmring4.html#K3" target="_self">Relocated</a> <font color="Maroon">p</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/>
<a NAME="E8:71"/><b>then </b>
<font color="Maroon">s</font> is <a href="ami_1.html#V9">halting</a>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:71"><i><font color="Green">E57</font></i></a>, <a class="ref" href="ami_1.html#D26">AMI_1:def 26</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:71"/><i><font color="Green">E80</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> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#D20">AMI_1:def 20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">t</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <a href="ami_2.html#K2">SCM-Data-Loc</a> </span>)</span> as   <a href="ami_1.html#NM4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="ami_5.html#T82">AMI_5:82</a></i>;<br/>
<a NAME="E12:71"/>
<font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s3</font>
 
;<br/>
<a NAME="E13:71"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="amistd_2.html#K8">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8">CurInstr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">t</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">u</font></span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="scmring4.html#E12"><i><font color="Green">Lemma58</font></i></a>, <a class="txt" href="scmring4.html#E13"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="scmring4.html#T8" target="_self">Th8</a>, <a class="txt" href="scmring4.html#E11"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">u</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D20">AMI_1:def 20</a><br/>

<b>thus </b><a NAME="E14:71"/>
 <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> <font color="Maroon">t</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K5">halt</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="amistd_2.html#T35">AMISTD_2:35</a></i>;<br/>


</div>
