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

<b>let </b><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<b>thus </b><a NAME="E2:60"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c1">b</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies  not <font color="Maroon" title="c1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:60_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:60_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>and </b><br/><a NAME="E3:60_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<a NAME="E4:60_1"/>
not <font color="Maroon" title="c1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E1:60"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="nat_3.html#E1:60_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="nat_3.html#E2:60_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:60_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E3:60_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/>


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

<b>assume </b><a NAME="E3:60"/><i><font color="Green" title="E30">A5</font></i>: 
not <font color="Maroon" title="c1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<a NAME="E4:60"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
not <font color="Maroon" title="c1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/>
<a NAME="E5:60"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_3.html#E3:60"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/>
<a NAME="E6:60"/>
1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/>
<a NAME="E7:60"/><b>then </b>
<font color="Maroon" title="c1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E8:60"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c1">b</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="nat_3.html#E1:60"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="nat_3.html#E4:60"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="nat_3.html#E5:60"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/>


</div>
