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

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





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</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="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:21_1_2"/>
not <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:21_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:21_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <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> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E5:21_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
