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

<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>thus </b><a NAME="E1:4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">i</font> implies ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">i</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E4:4_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1_1"/>
not <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<a NAME="E2:4_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:4_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E3:4"/><b>then </b>
 ex <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>


</div>
