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

<b>let </b><font color="Maroon" title="c1">i</font> be    <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> ;<br/>

<b>thus </b><a NAME="E1:12"/>
( <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Instr-Loc</a>  implies <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E10">A1</font></i>: 
<a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Instr-Loc</a> 
 ;<br/>

<b>assume </b><a NAME="E2:12_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:12_1"/><b>then </b>
(  ex <font color="Olive" title="b1">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>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  ex <font color="Olive" title="b1">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>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="scmpds_1.html#T14" target="_self" title="SCMPDS_1:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_1.html#E1:12_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="scmpds_1.html#D4" target="_self" title="SCMPDS_1:def.4">Def4</a>, <a class="ref" href="scmpds_1.html#T17" target="_self" title="SCMPDS_1:th.17">Th17</a>, <a class="ref" href="ami_2.html#T6" title="AMI_2:th.6">AMI_2:6</a></i>;<br/>


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

<b>thus </b><a NAME="E2:12"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <a href="scmpds_1.html#K2" title="SCMPDS_1:func.2">SCMPDS-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Instr-Loc</a>  )
 <b>by </b><i><a class="ref" href="scmpds_1.html#D4" target="_self" title="SCMPDS_1:def.4">Def4</a></i>;<br/>


</div>
