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

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="measure5.html#NM1">Interval</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>suppose </b></a><a NAME="E1:59_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:59_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_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="E1:59_1_1_1_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2"><b>suppose </b></a><a NAME="E1:59_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><a NAME="E2:59_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st ( <font color="Olive">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b1</font> or <font color="Olive">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2_1"/><i><font color="Green">E35</font></i>: 
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 <b>and </b><br/><a NAME="E2:59_1_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_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="E1:59_1_2_1_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_1_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="ref" href="measure5.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_1_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_1_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_1_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="txt" href="measure5.html#E1:59"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_1_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="txt" href="measure5.html#E2:59"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:59_1_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st ( <font color="Olive">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b1</font> or <font color="Olive">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2_2"/><i><font color="Green">E40</font></i>: 
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 <b>and </b><br/><a NAME="E2:59_1_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_2_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_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="E1:59_1_2_2_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_2_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_2_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a>, <a class="txt" href="measure5.html#E3:59"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_2_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_2_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_2_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_2_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:59_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st ( <font color="Olive">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b1</font> or <font color="Olive">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:59_1_2_3"/><i><font color="Green">E45</font></i>: 
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_3_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_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="E1:59_1_2_3_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_3_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i>, <a class="txt" href="measure5.html#E1:59_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_3_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:59_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> implies ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_2_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_4"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_4_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_4_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_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="E1:59_1_2_4_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_4_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i>, <a class="txt" href="measure5.html#E2:59_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_4_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_4_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i>, <a class="txt" href="measure5.html#E2:59_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_4_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_4_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_4_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_4_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_4_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_4_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_4_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_4_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:59_1_2"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st ( <font color="Olive">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b1</font> or <font color="Olive">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:59_1_2_5"/><i><font color="Green">E115</font></i>: 
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_5"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_5_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_5_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_5_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_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="E1:59_1_2_5_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_5_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_5_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_5_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_5_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_5_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_5_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_5_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_5_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_5_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_5_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="txt" href="measure5.html#E3:59_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_5_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_5_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:59_1_2"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st ( <font color="Olive">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b1</font> or <font color="Olive">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:59_1_2_6"/><i><font color="Green">E120</font></i>: 
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_6"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_6_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_6_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_6_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_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="E1:59_1_2_6_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_6_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_6_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_6_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_6_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_6_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_6_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_6_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_6_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_6_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_6_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_6_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_6_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_6_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:59_1_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> implies ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_2_7"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_7"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_7_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_7_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_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="E1:59_1_2_7_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_7_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i>, <a class="txt" href="measure5.html#E4:59_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_7_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_7_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i>, <a class="txt" href="measure5.html#E4:59_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_7_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_7_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_7_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_7_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_7_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_7_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_7_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_7_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_7_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_7_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_7_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E9:59_1_2"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st ( <font color="Olive">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b1</font> or <font color="Olive">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:59_1_2_8"/><i><font color="Green">E130</font></i>: 
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_8"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_8_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_8_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_8_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_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="E1:59_1_2_8_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_8_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_8_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_8_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_8_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_8_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_8_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_8_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_8_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_8_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_8_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_8_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_8_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a>, <a class="txt" href="measure5.html#E5:59_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_8_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_8_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:59_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> implies ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_2_9"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:59_1_2_9"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_9_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_9_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_9_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_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="E1:59_1_2_9_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> or <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_9_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_9_1_1_1_1_2"/>
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_9_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_9_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_9_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_9_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_9_1_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_9_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_2_9_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E11:59_1_2"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <b>by </b><i>, , <a class="ref" href="measure5.html#D1" target="_self">Def1</a>, , <a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a>, , <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, <a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
