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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></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">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E57</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1_1"><i><font color="Green">E57</font></i></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">E58</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1_1_1_1"><i><font color="Green">E58</font></i></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1"><i><font color="Green">E56</font></i></a></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">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1"><i><font color="Green">E56</font></i></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_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">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1_1_1_2"><i><font color="Green">E58</font></i></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1"><i><font color="Green">E56</font></i></a></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">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_1"><i><font color="Green">E56</font></i></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_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">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">E58</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">E59</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_1"><i><font color="Green">E59</font></i></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">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_1_1_1"><i><font color="Green">E60</font></i></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_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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="measure5.html#T22" target="_self">Th22</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="txt" href="measure5.html#E1:59_1_2_1"><i><font color="Green">E58</font></i></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">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="measure5.html#T24" target="_self">Th24</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="txt" href="measure5.html#E1:59_1_2_1"><i><font color="Green">E58</font></i></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">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_1_1_2"><i><font color="Green">E60</font></i></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_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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="measure5.html#T23" target="_self">Th23</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="txt" href="measure5.html#E1:59_1_2_1"><i><font color="Green">E58</font></i></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">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_1_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="measure5.html#T25" target="_self">Th25</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="txt" href="measure5.html#E1:59_1_2_1"><i><font color="Green">E58</font></i></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">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">E59</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">E60</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_2"><i><font color="Green">E60</font></i></a></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">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_2_1_1"><i><font color="Green">E61</font></i></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="measure5.html#T26" target="_self">Th26</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="txt" href="measure5.html#E1:59_1_2_2"><i><font color="Green">E59</font></i></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">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></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_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="measure5.html#T28" target="_self">Th28</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="txt" href="measure5.html#E1:59_1_2_2"><i><font color="Green">E59</font></i></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">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_2_1_2"><i><font color="Green">E61</font></i></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="measure5.html#T27" target="_self">Th27</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="txt" href="measure5.html#E1:59_1_2_2"><i><font color="Green">E59</font></i></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">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_2_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></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_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="measure5.html#T29" target="_self">Th29</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="txt" href="measure5.html#E1:59_1_2_2"><i><font color="Green">E59</font></i></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">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">E60</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_3"><i><font color="Green">E61</font></i></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">E62</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3_1_1"><i><font color="Green">E62</font></i></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T30" target="_self">Th30</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3"><i><font color="Green">E60</font></i></a></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">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></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_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3"><i><font color="Green">E60</font></i></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_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">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3_1_2"><i><font color="Green">E62</font></i></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3"><i><font color="Green">E60</font></i></a></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">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></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_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T33" target="_self">Th33</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_3"><i><font color="Green">E60</font></i></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_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">E60</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">E62</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_4"><i><font color="Green">E62</font></i></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">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_4_1_1"><i><font color="Green">E63</font></i></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T34" target="_self">Th34</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="txt" href="measure5.html#E1:59_1_2_4"><i><font color="Green">E61</font></i></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">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T34" target="_self">Th34</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="txt" href="measure5.html#E1:59_1_2_4"><i><font color="Green">E61</font></i></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">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_4_1_2"><i><font color="Green">E63</font></i></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_4_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_4"><i><font color="Green">E61</font></i></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">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_4_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_4_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_4"><i><font color="Green">E61</font></i></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">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">E62</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">E63</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_5"><i><font color="Green">E63</font></i></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">E64</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_5_1_1"><i><font color="Green">E64</font></i></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="measure5.html#T36" target="_self">Th36</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="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></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">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="measure5.html#T38" target="_self">Th38</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="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></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">E64</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_5_1_2"><i><font color="Green">E64</font></i></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="measure5.html#T37" target="_self">Th37</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="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></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">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_5_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="measure5.html#T39" target="_self">Th39</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="txt" href="measure5.html#E1:59_1_2_5"><i><font color="Green">E62</font></i></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">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">E63</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">E64</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_6"><i><font color="Green">E64</font></i></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">E65</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_6_1_1"><i><font color="Green">E65</font></i></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T40" target="_self">Th40</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="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></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">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T42" target="_self">Th42</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="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></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">E65</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_6_1_2"><i><font color="Green">E65</font></i></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T41" target="_self">Th41</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="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></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">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_6_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T43" target="_self">Th43</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="txt" href="measure5.html#E1:59_1_2_6"><i><font color="Green">E63</font></i></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">E63</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E64</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">E65</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_7"><i><font color="Green">E65</font></i></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">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_7_1_1"><i><font color="Green">E66</font></i></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_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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T44" target="_self">Th44</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="txt" href="measure5.html#E1:59_1_2_7"><i><font color="Green">E64</font></i></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">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T44" target="_self">Th44</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="txt" href="measure5.html#E1:59_1_2_7"><i><font color="Green">E64</font></i></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">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_7_1_2"><i><font color="Green">E66</font></i></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_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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T45" target="_self">Th45</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_7_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_7"><i><font color="Green">E64</font></i></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">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_7_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T45" target="_self">Th45</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_7_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_7"><i><font color="Green">E64</font></i></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">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">E65</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></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">E66</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_8"><i><font color="Green">E66</font></i></a></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">E67</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_8_1_1"><i><font color="Green">E67</font></i></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_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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></a>, <a class="ref" href="measure5.html#T46" target="_self">Th46</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="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></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">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></a>, <a class="ref" href="measure5.html#T48" target="_self">Th48</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="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></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">E67</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_8_1_2"><i><font color="Green">E67</font></i></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_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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></a>, <a class="ref" href="measure5.html#T47" target="_self">Th47</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="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></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">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_8_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></a>, <a class="ref" href="measure5.html#T49" target="_self">Th49</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="txt" href="measure5.html#E1:59_1_2_8"><i><font color="Green">E65</font></i></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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E65</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">E66</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:59_1_2_9"><i><font color="Green">E66</font></i></a></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">E67</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_9_1_1"><i><font color="Green">E67</font></i></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_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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_9"><i><font color="Green">E65</font></i></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">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_9"><i><font color="Green">E65</font></i></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">E67</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_9_1_2"><i><font color="Green">E67</font></i></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_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">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_9"><i><font color="Green">E65</font></i></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">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_9_1_2_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( not <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="ref" href="measure5.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E3:59_1_2_9_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:59_1_2_9"><i><font color="Green">E65</font></i></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:59"><i><font color="Green">E53</font></i></a>, <a class="txt" href="measure5.html#E2:59"><i><font color="Green">E54</font></i></a>, <a class="txt" href="measure5.html#E3:59"><i><font color="Green">E55</font></i></a>, <a class="txt" href="measure5.html#E1:59_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="measure5.html#E2:59_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E3:59_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="measure5.html#E4:59_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="measure5.html#E5:59_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E6:59_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="measure5.html#E7:59_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E8:59_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="measure5.html#E9:59_1_2"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
