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

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



<b>assume </b><a NAME="E1:76"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1"/>
(  ex <font color="Olive">j1</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">j1</font> &amp; <font color="Olive">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> ) or ( <font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><b>suppose </b></a><a NAME="E1:76_1_1"/>
 ex <font color="Olive">j1</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">j1</font> &amp; <font color="Olive">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E4:76_1_1"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><b>assume </b><a NAME="E5:76_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:76_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:76_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:76_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2"><b>suppose </b></a><a NAME="E1:76_1_2"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
