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<b>let </b><font color="Maroon">IT1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">IT2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><font color="Maroon">IT1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> ) )
 <b>and </b><br/><a NAME="E2:45_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">IT2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><font color="Maroon">IT2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:45_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">IT1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1"/>
( <font color="Maroon">IT1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">IT2</font> or <font color="Maroon">IT1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">IT2</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="45_1_2_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1"/>
<font color="Maroon">IT1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">IT2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_2"/>
<font color="Maroon">IT1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">IT2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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