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

<b>set </b><font color="Maroon">b</font> =  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>();<br/>
<b>set </b><font color="Maroon">FR</font> =  <a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>();<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>   st <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">t</font>] holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Olive">t</font></span>)</span>] );<br/>
<a NAME="E1:19_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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 <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>

<b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">t</font>] )
 ;<br/>

<a NAME="E2:19_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="scmpds_2.html#M1">Int_position</a>  holds  <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="scmpds_7.html#T66">SCMPDS_7:66</a></i>;<br/>
<b>thus </b><a NAME="E3:19_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, <a class="ref" href="scmpds_7.html#T66">SCMPDS_7:66</a></i>;<br/>

<b>thus </b><a NAME="E4:19_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">t</font></span>)</span>]
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">u</font>] )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_2_1_1_1"/>
<font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">u</font></span>)</span>] )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:19_1_2_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Maroon">u</font>]
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">Ad</font> =  <a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span>;<br/><a NAME="E4:19_1_2_1_1_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">t</font> being  <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a>   st <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <font color="Olive">t</font>] &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Olive">t</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Olive">t</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Olive">t</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Olive">t</font></span>)</span>] )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">Iu</font> =  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font>;<br/><a NAME="E5:19_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">u</font>] or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">u</font>] ) &amp;  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_4.html#K9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> )
 <b>from </b><i><a class="ref" href="scpisort.html#S2" target="_self">SCPISORT:sch 2</a>(, , , );<br/></i><a NAME="E6:19_1_2_1_1_2"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>5</sub></font>()
 <b>by </b><i>, , </i>;<br/><a NAME="E7:19_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">F<sub>5</sub></font>() <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E8:19_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span>
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E9:19_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:19_1_2_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:19_1_2_1_1_2"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i>, , </i>;<br/><a NAME="E12:19_1_2_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">F<sub>5</sub></font>()</span>)</span></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span>]
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E13:19_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">u</font></span>)</span>] )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:19_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:19_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>thus </b><a NAME="E4:19_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>()] or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>()] )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_3"/>
( <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_1"><b>suppose </b></a><a NAME="E1:19_1_3_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:19_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:19_1_3_1"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_3_2"><b>suppose </b></a><a NAME="E1:19_1_3_2"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_3_2"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="scmpds_2.html#K2">.</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="scmpds_8.html#K1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_7.html#K2">for-down</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">F<sub>1</sub></font>()</span>)</span>] )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
