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

<b>take </b><font color="Maroon">H</font> =  <a href="scmpds_2.html#K4" target="_self">goto</a> 0; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D9">AMI_1:def 9</a><br/>

<b>thus </b><a NAME="E1:103_1_1"/>
<font color="Maroon">H</font> is <a href="ami_1.html#V3">halting</a>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">W</font> be   <a href="ami_1.html#NM3">Instruction</a> of <a href="scmpds_2.html#K1" target="_self">SCMPDS</a> ;<br/>

<b>assume </b><a NAME="E2:103_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">W</font> is <a href="ami_1.html#V3">halting</a>
 ;<br/>

<b>assume </b><a NAME="E3:103_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_1_1_1"/>
(  ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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">W</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> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_1"><b>suppose </b></a><a NAME="E1:103_1_1_1_1"/>
 ex <font color="Olive">k1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Olive">k1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:103_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K4" target="_self">goto</a> <font color="Maroon">k1</font>
 ;<br/><b>thus </b><a NAME="E4:103_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_2"><b>suppose </b></a><a NAME="E1:103_1_1_1_2"/>
 ex <font color="Olive">a</font> being   <a href="scmpds_2.html#M1" target="_self">Int_position</a>  st <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="scmpds_2.html#K5" target="_self">return</a> <font color="Olive">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_3"><b>suppose </b></a><a NAME="E1:103_1_1_1_3"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_4"><b>suppose </b></a><a NAME="E1:103_1_1_1_4"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_5"><b>suppose </b></a><a NAME="E1:103_1_1_1_5"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_6"><b>suppose </b></a><a NAME="E1:103_1_1_1_6"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_7"><b>suppose </b></a><a NAME="E1:103_1_1_1_7"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_8"><b>suppose </b></a><a NAME="E1:103_1_1_1_8"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_9"><b>suppose </b></a><a NAME="E1:103_1_1_1_9"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_10"><b>suppose </b></a><a NAME="E1:103_1_1_1_10"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_11"><b>suppose </b></a><a NAME="E1:103_1_1_1_11"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_12"><b>suppose </b></a><a NAME="E1:103_1_1_1_12"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_12"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_13"><b>suppose </b></a><a NAME="E1:103_1_1_1_13"/>
 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">W</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:103_1_1_1_13"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1_14"><b>suppose </b></a><a NAME="E1:103_1_1_1_14"/>
 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">W</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 class="add"><b>hence </b><a NAME="E2:103_1_1_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_2.html#E17"><i><font color="Green">Lemma60</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
