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

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_2_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:113_1_2_1"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="newton.html#K12">!</a>  )
 <b>and </b><br/><a NAME="E2:113_1_2_1"/><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="newton.html#K12">!</a>  )
 ;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:113_1_2_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="newton.html#K12">!</a>  )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:113_1_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="newton.html#K12">!</a>  )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:113_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><b>thus </b><a NAME="E2:113_1_2_1_1"/>
contradiction
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1_2_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:113_1_2_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:113_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:113_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:113_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:113_1_2_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:113_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E3:113_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="newton.html#K12">!</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:113_1_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:113_1_2_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:113_1_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="newton.html#K12">!</a>  ) &amp;  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a>  &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="newton.html#K12">!</a>  ) implies <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> ) )
 ;<br/>


</div>
