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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:12"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> implies <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</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"/>
<font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E4:12_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:12"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E4:12_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</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>hence </b><a NAME="E4:12"/>
( <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> iff <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>


</div>
