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

<b>let </b><font color="Maroon">s</font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D8">AMI_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">s</font> as   <a href="scmpds_1.html#NM1">SCMPDS-State</a> ;<br/>
<b>set </b><font color="Maroon">I</font> =  <a href="scmpds_2.html#K4" target="_self">goto</a> 0;<br/>
<i><font color="Green">E22</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">S</font> = 
 <a href="relat_1.html#K1">dom</a> <a href="scmpds_1.html#K2">SCMPDS-OK</a> 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">Es</font> =  <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" target="_self">goto</a> 0</span>)</span>,<font color="Maroon">s</font> as   <a href="scmpds_1.html#NM1">SCMPDS-State</a> ;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:86_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 </span><b>by </b><i>, , <a class="ref" href="scmpds_1.html#T14">SCMPDS_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_1"><b>suppose </b></a><a NAME="E1:86_2_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:86_2_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> &amp;  <a href="scmpds_2.html#K19" target="_self">ICplusConst</a> <font color="Maroon">s</font>,0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">n</font> =  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  ;<br/><a NAME="E5:86_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="ami_2.html#D3">AMI_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:86_2_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>consider </b><font color="Maroon">n0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:86_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n0</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">n0</font> = <font color="Maroon">n0</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E11:86_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n0</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E12:86_2_1_1"/><i><font color="Green">E65</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><b>thus </b><font color="Maroon">Es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n0</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_2"><b>suppose </b></a><a NAME="E1:86_2_1_2"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:86_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 <b>by </b><i><a class="ref" href="ami_2.html#D2">AMI_2:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</font> as    <a href="scmpds_2.html#M1" target="_self">Int_position</a>  <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">Es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">s</font> <a href="scmpds_2.html#K2" target="_self">.</a> <font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_3"><b>suppose </b></a><a NAME="E1:86_2_1_3"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:86_2_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E4:86_2_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:86_2_1_3"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E6:86_2_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 <b>by </b><i>, <a class="ref" href="ami_2.html#D3">AMI_2:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  ;<br/><b>reconsider </b><font color="Maroon">I0</font> =  <a href="scmpds_2.html#K4" target="_self">goto</a> 0 as    <a href="subset_1.html#M1">Element</a> of  <a href="scmpds_1.html#K1">SCMPDS-Instr</a>  ;<br/><b>reconsider </b><font color="Maroon">i</font> = 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14 <b>by </b><i><a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/> <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" target="_self">goto</a> 0</span>)</span>,<font color="Maroon">s</font> = 
 <a href="scmpds_1.html#K24">SCM-Exec-Res</a> <font color="Maroon">I0</font>,<font color="Maroon">S</font>
<b>by </b><i/>
<br/>.= 
 <a href="scmpds_1.html#K4">SCM-Chg</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="scmpds_1.html#K8">jump_address</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">I0</font> <a href="scmpds_1.html#K11">const_INT</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="scmpds_1.html#D24">SCMPDS_1:def 24</a></i>
;<br/><b>hence </b><font color="Maroon">Es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="scmpds_1.html#T28">SCMPDS_1:28</a></i>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>

;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Es</font> = 
 <a href="relat_1.html#K1">dom</a> <a href="scmpds_1.html#K2">SCMPDS-OK</a> 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:86"/>
 <a href="ami_1.html#K4">Exec</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K4" target="_self">goto</a> 0</span>)</span>,<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
