<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="ami_1.html#V15" title="AMI_1:attr.15">programmed</a> <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ;<br/>



<a NAME="E1:41"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#D6" target="_self" title="SCMFSA_4:def.6">Def6</a></i>;<br/>
<a NAME="E2:41"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="scmfsa_4.html#D6" target="_self" title="SCMFSA_4:def.6">Def6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E28">A2</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span> = 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  :  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> </span>}</span> 
<b>by </b><i><a class="ref" href="scmfsa_4.html#D7" target="_self" title="SCMFSA_4:def.7">Def7</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_4.html#D7" target="_self" title="SCMFSA_4:def.7">Def7</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:41_2"/><i><font color="Green" title="E29">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="ami_1.html#D40" title="AMI_1:def.40">AMI_1:def 40</a></i>;<br/><b>assume </b><a NAME="E2:41_2"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of  <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <b>by </b><i><a class="txt" href="scmfsa_4.html#E1:41_2"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/><a NAME="E4:41_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  :  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E2:41_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="scmfsa_4.html#D7" target="_self" title="SCMFSA_4:def.7">Def7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">m</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> <b> such that </b><br/><a NAME="E6:41_2"/><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span> &amp;  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> )
 ;<br/><a NAME="E7:41_2"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E6:41_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="scmfsa_4.html#D1" target="_self" title="SCMFSA_4:def.1">Def1</a></i>;<br/><a NAME="E8:41_2"/><i><font color="Green" title="E33">A7</font></i>: 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E6:41_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="scmfsa_4.html#D6" target="_self" title="SCMFSA_4:def.6">Def6</a></i>;<br/><a NAME="E9:41_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">mm</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">mm</font> is    <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>  :  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">mm</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">p</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="scmfsa_4.html#D7" target="_self" title="SCMFSA_4:def.7">Def7</a></i>;<br/><a NAME="E10:41_2"/><b>then </b><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c5">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E6:41_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="scmfsa_4.html#E8:41_2"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/><i><font color="Green" title="E35">A9</font></i>:  <a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font></span>)</span> = 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E8:41_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="ami_1.html#D47" title="AMI_1:def.47">AMI_1:def 47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">+</a> <font color="Maroon" title="c2">j</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E8:41_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="scmfsa_4.html#T30" target="_self" title="SCMFSA_4:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmfsa_4.html#D1" target="_self" title="SCMFSA_4:def.1">Def1</a></i>
<br/>.= 
 <a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c5">x'</font>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E6:41_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="scmfsa_4.html#E10:41_2"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="ami_1.html#D47" title="AMI_1:def.47">AMI_1:def 47</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E6:41_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="scmfsa_4.html#E7:41_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="scmfsa_4.html#T30" target="_self" title="SCMFSA_4:th.30">Th30</a></i>
<br/>.= 
 <a href="scmfsa_4.html#K3" title="SCMFSA_4:func.3">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">pi</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c5">x'</font></span>)</span>,<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E8:41_2"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="scmfsa_4.html#E11:41_2"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="scmfsa_4.html#T24" target="_self" title="SCMFSA_4:th.24">Th24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="scmfsa_4.html#E10:41_2"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="scmfsa_4.html#T24" target="_self" title="SCMFSA_4:th.24">Th24</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
 <a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_4.html#K5" title="SCMFSA_4:func.5">IncAddr</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K6" title="SCMFSA_4:func.6">Shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">j</font></span>)</span>,<font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="scmfsa_4.html#E3:41"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
