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

<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">aa</font>, <font color="Maroon" title="c3">bb</font> be    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="scmfsa_2.html#M2" title="SCMFSA_2:mode.2">FinSeq-Location</a> ;<br/>







<b>set </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">aa</font>;<br/>
<b>set </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c3">bb</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E45">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font> &amp; <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font> &amp; <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E3:54"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">sa</font> = <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font> as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E5:54"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">sb</font> = <font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font> as    <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>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E7:54"/><i><font color="Green" title="E47">A3</font></i>: 
( <font color="Maroon" title="c7">sa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c8">sb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:54"/><i><font color="Green" title="E48">A4</font></i>: 
<span class="p1">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p2">(<span class="default"><a href="sfmastr3.html#K5" title="SFMASTR3:func.5">swap</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">aa</font>,<font color="Maroon" title="c3">bb</font></span>)</span>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sfmastr3.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="sfmastr3.html#E2:54"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="sfmastr3.html#T39" target="_self" title="SFMASTR3:th.39">Th39</a></i>;<br/>
<a NAME="E9:54"/><i><font color="Green" title="E49">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( <font color="Maroon" title="c7">sa</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">sb</font> or <font color="Maroon" title="c7">sa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">sb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c7">sa</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">sb</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p3">(<span class="default"><a href="sfmastr3.html#K5" title="SFMASTR3:func.5">swap</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">aa</font>,<font color="Maroon" title="c3">bb</font></span>)</span>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span>
<b>by </b><i><a class="txt" href="sfmastr3.html#E8:54"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span>
<b>by </b><i><a class="txt" href="sfmastr3.html#E7:54"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>
;<br/><br/><b>thus </b><a NAME="E3:54_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p3">(<span class="default"><a href="sfmastr3.html#K5" title="SFMASTR3:func.5">swap</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">aa</font>,<font color="Maroon" title="c3">bb</font></span>)</span>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E7:54"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E8:54"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E9:54"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/>
<font color="Maroon" title="c7">sa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">sb</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p3">(<span class="default"><a href="sfmastr3.html#K5" title="SFMASTR3:func.5">swap</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">aa</font>,<font color="Maroon" title="c3">bb</font></span>)</span>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E7:54"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E8:54"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E9:54"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>thus </b><a NAME="E3:54_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa6b.html#K3" title="SCMFSA6B:func.3">IExec</a> <span class="p3">(<span class="default"><a href="sfmastr3.html#K5" title="SFMASTR3:func.5">swap</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">aa</font>,<font color="Maroon" title="c3">bb</font></span>)</span>,<font color="Maroon" title="c1">s</font></span>)</span> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c3">bb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K21" title="SCMFSA_2:func.21">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="scmfsa_2.html#K20" title="SCMFSA_2:func.20">.</a> <font color="Maroon" title="c2">aa</font></span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr3.html#E7:54"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="sfmastr3.html#E8:54"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="sfmastr3.html#E9:54"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
