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

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ;<br/><b>let </b><font color="Maroon" title="c2">l</font> be    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of  <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">i</font> =  <a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 </span>}</span> ;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">l</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">Following</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <b>and </b><br/><a NAME="E4:19_1"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font>
 <b>and </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font>
 ;<br/><i><font color="Green" title="E19">A4</font></i>: <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E3:19_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scmpds_9.html#E4:19_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="scmpds_9.html#E5:19_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="scmpds_9.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
<b>by </b><i><a class="ref" href="scmpds_1.html#D23" title="SCMPDS_1:def.23">SCMPDS_1:def 23</a>, <a class="ref" href="scmpds_2.html#T70" title="SCMPDS_2:th.70">SCMPDS_2:70</a></i>
;<br/><a NAME="E7:19_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</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>  : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="scmpds_9.html#E3:19_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="ami_2.html#D2" title="AMI_2:def.2">AMI_2:def 2</a>, <a class="ref" href="scmpds_2.html#D1" title="SCMPDS_2:def.1">SCMPDS_2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">k</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="E9:19_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><a NAME="E10:19_1"/>
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:19_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scmpds_9.html#E6:19_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="scmpds_9.html#E9:19_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/><a NAME="E12:19_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E13:19_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 </span>}</span> 
 <b>by </b><i><a class="txt" href="scmpds_9.html#E9:19_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">k</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="E4:19"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">k2</font> = <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 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="txt" href="scmpds_9.html#E5:19"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="scmpds_9.html#E17"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/>
<a NAME="E7:19"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="scmpds_2.html#D2" title="SCMPDS_2:def.2">SCMPDS_2:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">j</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="E9:19"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="ami_2.html#T32" title="AMI_2:th.32">AMI_2:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">s</font> being   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">il1</font> = <font color="Maroon" title="c2">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <b>by </b><i><a class="ref" href="ami_1.html#D11" title="AMI_1:def.11">AMI_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">I</font> =  <a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c2">l</font> <b>by </b><i><a class="ref" href="ami_1.html#D14" title="AMI_1:def.14">AMI_1:def 14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">t</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E13:19"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="ami_2.html#T33" title="AMI_2:th.33">AMI_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">t1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  <b>by </b><i><a class="ref" href="scmpds_2.html#T9" title="SCMPDS_2:th.9">SCMPDS_2:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font>;<br/>
<b>set </b><font color="Maroon" title="c16">v</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span>;<br/>
<a NAME="E15:19"/><i><font color="Green" title="E19">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p3">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E16:19"/><i><font color="Green" title="E20">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="ami_6.html#T6" title="AMI_6:th.6">AMI_6:6</a></i>;<br/>
<i><font color="Green" title="E21">A11</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c2">l</font>
<b>by </b><i><a class="ref" href="ami_6.html#T6" title="AMI_6:th.6">AMI_6:6</a></i>
;<br/>
<a NAME="E18:19"/>
( <font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t1</font> )
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T53" title="SCMPDS_2:th.53">SCMPDS_2:53</a></i>;<br/>
<a NAME="E19:19"/><b>then </b>
not <font color="Maroon" title="c2">l</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"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E15:19"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E20:19"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E16:19"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/>
<a NAME="E21:19"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  &amp; <font color="Maroon" title="c14">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  )
 
<b>by </b><i><a class="ref" href="scmpds_2.html#T52" title="SCMPDS_2:th.52">SCMPDS_2:52</a></i>;<br/>
<a NAME="E22:19"/><b>then </b><i><font color="Green" title="E23">A13</font></i>: 
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  <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"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E15:19"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<i><font color="Green" title="E24">A14</font></i>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c2">l</font>
<b>by </b><i><a class="txt" href="scmpds_9.html#E17:19"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="scmpds_9.html#E22:19"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
;<br/>
<a NAME="E24:19"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E25:19"/><b>then </b><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E9:19"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E26:19"/><b>then </b><i><font color="Green" title="E26">A16</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T89" title="FUNCT_4:th.89">FUNCT_4:89</a></i>;<br/>
<a NAME="E27:19"/><i><font color="Green" title="E27">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k2</font>
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E25:19"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="funct_4.html#T89" title="FUNCT_4:th.89">FUNCT_4:89</a></i>;<br/>
<i><font color="Green" title="E28">A18</font></i>:  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c8">j</font>,1 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="scmpds_2.html#D4" title="SCMPDS_2:def.4">SCMPDS_2:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c17">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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="txt" href="scmpds_9.html#E25:19"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="funct_4.html#T89" title="FUNCT_4:th.89">FUNCT_4:89</a></i>;<br/>
<font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_9.html#E4:19"><i><font color="Green" title="E16">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2

<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c17">r</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_9.html#E27:19"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c8">j</font>,1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
<b>by </b><i><a class="txt" href="scmpds_9.html#E28:19"><i><font color="Green" title="E28">A18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</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"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E26:19"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="scmpds_1.html#D23" title="SCMPDS_1:def.23">SCMPDS_1:def 23</a>, <a class="ref" href="scmpds_2.html#T70" title="SCMPDS_2:th.70">SCMPDS_2:70</a></i>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">Following</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="ami_1.html#K15" title="AMI_1:func.15">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c11">I</font></span>)</span></span>)</span> <a href="ami_1.html#K33" title="AMI_1:func.33">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c14">t1</font> <a href="scmpds_9.html#K1" title="SCMPDS_9:func.1">--&gt;</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c7">k2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmpds_9.html#E20:19"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="scmpds_9.html#E23:19"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="scmpds_9.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E31:19"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K5" title="SCMPDS_2:func.5">return</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="scmpds_9.html#E20:19"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="scmpds_9.html#E23:19"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/>


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