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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="scmpds_4.html#K1">Load</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:80"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1"/>
( <font color="Maroon">c<sub>3</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>3</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</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="80_1_1"><b>suppose </b></a><a NAME="E1:80_1_1"/>
<font color="Maroon">c<sub>3</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_1_1"/>
(  <a href="scmpds_7.html#K2" target="_self">for-down</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1,<span class="p1">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="scmpds_7.html#K2" target="_self">for-down</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1,<span class="p1">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="scmpds_7.html#T63" target="_self">Th63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2"><b>suppose </b></a><a NAME="E1:80_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:80_1_2"/>
 <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2 <a href="hidden.html#R1">=</a>  <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_7.html#E1:80"><i><font color="Green">E60</font></i></a>, <a class="ref" href="scmp_gcd.html#T5">SCMP_GCD:5</a></i>;<br/><a NAME="E3:80_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmp_gcd.html#K2">GBP</a> 
 <b>by </b><i><a class="ref" href="scmp_gcd.html#T4">SCMP_GCD:4</a>, <a class="ref" href="scmp_gcd.html#D2">SCMP_GCD:def 2</a></i>;<br/><a NAME="E4:80_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
not  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="scmp_gcd.html#K2">GBP</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:80_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="scmpds_5.html#T6">SCMPDS_5:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></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="E1:80_1_2_1"/><i><font color="Green">E65</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="scmpds_2.html#M1">Int_position</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="scmp_gcd.html#K2">GBP</a> </span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  )
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E2:80_1_2_1"/>
<span class="p1">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="scmpds_7.html#E1:80"><i><font color="Green">E60</font></i></a>, <a class="txt" href="scmpds_7.html#E1:80_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="scmpds_5.html#T40">SCMPDS_5:40</a></i>;<br/><a NAME="E3:80_1_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,3 <a href="hidden.html#R1">=</a>  <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 3</span>)</span>
 <b>by </b><i><a class="ref" href="scmp_gcd.html#T5">SCMP_GCD:5</a></i>;<br/><a NAME="E4:80_1_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmp_gcd.html#K2">GBP</a> 
 <b>by </b><i><a class="ref" href="scmp_gcd.html#T4">SCMP_GCD:4</a>, <a class="ref" href="scmp_gcd.html#D2">SCMP_GCD:def 2</a></i>;<br/><b>thus </b><i><font color="Green">E68</font></i>: <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> 
<b>by </b><i><a class="ref" href="scmpds_5.html#T45">SCMPDS_5:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> 
<b>by </b><i><a class="txt" href="scmpds_7.html#E4:80_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="scmpds_2.html#T60">SCMPDS_2:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> 
<b>by </b><i><a class="ref" href="scmpds_5.html#T40">SCMPDS_5:40</a></i>
;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2;<br/><a NAME="E6:80_1_2_1"/>
 <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2 <a href="hidden.html#R1">=</a>  <a href="scmp_gcd.html#K1">intpos</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="scmpds_7.html#E1:80"><i><font color="Green">E60</font></i></a>, <a class="ref" href="scmp_gcd.html#T5">SCMP_GCD:5</a></i>;<br/><a NAME="E7:80_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="scmpds_2.html#K3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2
 <b>by </b><i><a class="txt" href="scmpds_7.html#E3:80_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="scmp_gcd.html#T4">SCMP_GCD:4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>5</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span>
<b>by </b><i><a class="ref" href="scmpds_5.html#T45">SCMPDS_5:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_5.html#K1">Initialized</a> <font color="Maroon">c<sub>5</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span>
<b>by </b><i><a class="txt" href="scmpds_7.html#E7:80_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="scmpds_2.html#T60">SCMPDS_2:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</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">c<sub>3</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a> </span>)</span>,2</span>)</span>
<b>by </b><i><a class="ref" href="scmpds_5.html#T40">SCMPDS_5:40</a></i>
;<br/><b>thus </b><a NAME="E9:80_1_2_1"/>
(  <a href="scmpds_4.html#K1">Load</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="scmpds_4.html#K1">Load</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="scmpds_6.html#T34">SCMPDS_6:34</a>, <a class="ref" href="scmpds_6.html#T35">SCMPDS_6:35</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="scmpds_2.html#M1">Int_position</a> ;<br/><b>assume </b><a NAME="E1:80_1_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="scmp_gcd.html#K2">GBP</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:80_1_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmp_gcd.html#K2">GBP</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:80_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="scmpds_2.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="scmpds_7.html#E5:80_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:80_1_2"/>
(  <a href="scmpds_7.html#K2" target="_self">for-down</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1,<span class="p1">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="scmpds_7.html#K2" target="_self">for-down</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1,<span class="p1">(<span class="default"><a href="scmpds_4.html#K1">Load</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="scmpds_7.html#E1:80_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="scmpds_7.html#E3:80_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="scmpds_7.html#E4:80_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="scmpds_7.html#E5:80_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="scmpds_7.html#T67" target="_self">Th67</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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