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

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:102"/>
(  not <font color="Maroon">I</font> is   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  or  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:102_1"/>
<font color="Maroon">I</font> is   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1_1"/>
( <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : verum </span>}</span>  or <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">d1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">d1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span>  or <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I1</font>,<span class="p3"><a href="scmpds_1.html#K9">&lt;*</a><span class="default"><font color="Olive">d2</font>,<font color="Olive">k2</font></span><a href="scmpds_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Olive">d2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">k2</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">I1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,3</span><a href="tarski.html#K2">}</a></span> </span>}</span>  or <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I2</font>,<span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Olive">d3</font>,<font color="Olive">k3</font>,<font color="Olive">k4</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Olive">d3</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">k3</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , <font color="Olive">k4</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">I2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">4,5,6,7,8</span><a href="enumset1.html#K3">}</a></span> </span>}</span>  or <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I3</font>,<span class="p3"><a href="finseq_4.html#K8">&lt;*</a><span class="default"><font color="Olive">d4</font>,<font color="Olive">d5</font>,<font color="Olive">k5</font>,<font color="Olive">k6</font></span><a href="finseq_4.html#K8">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I3</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Olive">d4</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">d5</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">k5</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , <font color="Olive">k6</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">I3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">9,10,11,12,13</span><a href="enumset1.html#K3">}</a></span> </span>}</span>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_1"><b>suppose </b></a><a NAME="E1:102_1_1_1"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">k1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:102_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">k1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E4:102_1_1_1"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a></i>;<br/><b>hence </b><a NAME="E5:102_1_1_1"/>
(  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_2"><b>suppose </b></a><a NAME="E1:102_1_1_2"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">d1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">d1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a>  : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> <b> such that </b><br/><a NAME="E3:102_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">d1</font> as    <a href="scmpds_2.html#M1" target="_self">Int_position</a>  <b>by </b><i/>;<br/><a NAME="E5:102_1_1_2"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a></i>;<br/><b>hence </b><a NAME="E6:102_1_1_2"/>
(  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_3"><b>suppose </b></a><a NAME="E1:102_1_1_3"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I1</font>,<span class="p3"><a href="scmpds_1.html#K9">&lt;*</a><span class="default"><font color="Olive">d2</font>,<font color="Olive">k2</font></span><a href="scmpds_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Olive">d2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">k2</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">I1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,3</span><a href="tarski.html#K2">}</a></span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">I2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Maroon">d2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:102_1_1_3"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">I2</font>,<span class="p2"><a href="scmpds_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">d2</font>,<font color="Maroon">k2</font></span><a href="scmpds_1.html#K9">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">I2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">2,3</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">d2</font> as    <a href="scmpds_2.html#M1" target="_self">Int_position</a>  <b>by </b><i/>;<br/><a NAME="E5:102_1_1_3"/>
( <font color="Maroon">I2</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">I2</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:102_1_1_3"/><b>then </b>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Maroon">a</font>,<font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Maroon">k2</font> )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a></i>;<br/><b>hence </b><a NAME="E7:102_1_1_3"/>
(  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_4"><b>suppose </b></a><a NAME="E1:102_1_1_4"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I2</font>,<span class="p3"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Olive">d3</font>,<font color="Olive">k3</font>,<font color="Olive">k4</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Olive">d3</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">k3</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , <font color="Olive">k4</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">I2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">4,5,6,7,8</span><a href="enumset1.html#K3">}</a></span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">I3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Maroon">d3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:102_1_1_4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">I3</font>,<span class="p2"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">d3</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="finseq_1.html#K11">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">I3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">4,5,6,7,8</span><a href="enumset1.html#K3">}</a></span> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">d3</font> as    <a href="scmpds_2.html#M1" target="_self">Int_position</a>  <b>by </b><i/>;<br/><a NAME="E5:102_1_1_4"/>
( <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 5 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 6 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 7 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 8 )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><a NAME="E6:102_1_1_4"/><b>then </b>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font> )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a></i>;<br/><b>hence </b><a NAME="E7:102_1_1_4"/>
(  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1_5"><b>suppose </b></a><a NAME="E1:102_1_1_5"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">I3</font>,<span class="p3"><a href="finseq_4.html#K8">&lt;*</a><span class="default"><font color="Olive">d4</font>,<font color="Olive">d5</font>,<font color="Olive">k5</font>,<font color="Olive">k6</font></span><a href="finseq_4.html#K8">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">I3</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Olive">d4</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">d5</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Olive">k5</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , <font color="Olive">k6</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  : <font color="Olive">I3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">9,10,11,12,13</span><a href="enumset1.html#K3">}</a></span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">I3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 14, <font color="Maroon">d4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Maroon">d5</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K2">SCM-Data-Loc</a> , <font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:102_1_1_5"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">I3</font>,<span class="p2"><a href="finseq_4.html#K8">&lt;*</a><span class="default"><font color="Maroon">d4</font>,<font color="Maroon">d5</font>,<font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="finseq_4.html#K8">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">I3</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">9,10,11,12,13</span><a href="enumset1.html#K3">}</a></span> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">d4</font>, <font color="Maroon">b</font> = <font color="Maroon">d5</font> as    <a href="scmpds_2.html#M1" target="_self">Int_position</a>  <b>by </b><i/>;<br/><a NAME="E5:102_1_1_5"/>
( <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 9 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 10 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 11 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 12 or <font color="Maroon">I3</font> <a href="hidden.html#R1">=</a> 13 )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="enumset1.html#D3">ENUMSET1:def 3</a></i>;<br/><a NAME="E6:102_1_1_5"/><b>then </b>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font>,<font color="Maroon">b</font>,<font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font>,<font color="Maroon">b</font>,<font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font>,<font color="Maroon">b</font>,<font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font>,<font color="Maroon">b</font>,<font color="Maroon">k2</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<font color="Maroon">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Maroon">b</font>,<font color="Maroon">k2</font> )
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a></i>;<br/><b>hence </b><a NAME="E7:102_1_1_5"/>
(  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:102"/>
( (  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K7" target="_self">saveIC</a> <font color="Olive">a</font>,<font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="scmpds_2.html#K6" target="_self">:=</a> <font color="Olive">k1</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K11" target="_self">:=</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K8" target="_self">&lt;&gt;0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K9" target="_self">&lt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K10" target="_self">&gt;=0_goto</a> <font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K12" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K13" target="_self">AddTo</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K14" target="_self">SubFrom</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K15" target="_self">MultBy</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K16" target="_self">Divide</a> <font color="Olive">a</font>,<font color="Olive">k1</font>,<font color="Olive">b</font>,<font color="Olive">k2</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  ex <font color="Olive">k1</font>, <font color="Olive">k2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>,<font color="Olive">k1</font> <a href="scmpds_2.html#K17" target="_self">:=</a> <font color="Olive">b</font>,<font color="Olive">k2</font> ) implies <font color="Maroon">I</font> is   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a>  )
 ;<br/>


</div>
