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

<b>let </b><font color="Maroon" title="c1">a1</font>, <font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1"/>
( ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>suppose </b></a><a NAME="E1:61_1_1"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:61_1_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_1_1"/><i><font color="Green" title="E57">A5</font></i>: 
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_1_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1_1"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_1"/><i><font color="Green" title="E58">A6</font></i>: 
<font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1_1_1_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T20" target="_self" title="MEASURE5:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T20" target="_self" title="MEASURE5:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1"><i><font color="Green" title="E56">A4</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="E3:61_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_2"/><i><font color="Green" title="E58">A7</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font> or <font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1_1_1_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T21" target="_self" title="MEASURE5:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T21" target="_self" title="MEASURE5:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_1"><i><font color="Green" title="E56">A4</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="E3:61_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="61_1_2"><b>suppose </b></a><a NAME="E1:61_1_2"/><i><font color="Green" title="E56">A8</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c5">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 ;<br/><div class="add"><a NAME="E2:61_1_2"/><i><font color="Green" title="E57">A9</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b1</font> or <font color="Olive" title="b2">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Olive" title="b2">a2</font>,<font color="Olive" title="b4">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:61_1_2_1"/><i><font color="Green" title="E58">A10</font></i>: 
( ( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 <b>and </b><br/><a NAME="E2:61_1_2_1"/><i><font color="Green" title="E59">A11</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_1"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_1"/><i><font color="Green" title="E60">A12</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1_1_1"><i><font color="Green" title="E60">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="measure5.html#T22" target="_self" title="MEASURE5:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="measure5.html#T24" target="_self" title="MEASURE5:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</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="E3:61_1_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_2"/><i><font color="Green" title="E60">A13</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1_1_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_1_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="measure5.html#T23" target="_self" title="MEASURE5:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="measure5.html#T25" target="_self" title="MEASURE5:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_1"><i><font color="Green" title="E58">A10</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="E3:61_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:61_1_2"/><i><font color="Green" title="E58">A14</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b1</font> or <font color="Olive" title="b2">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Olive" title="b2">a2</font>,<font color="Olive" title="b4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:61_1_2_2"/><i><font color="Green" title="E59">A15</font></i>: 
( ( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 <b>and </b><br/><a NAME="E2:61_1_2_2"/><i><font color="Green" title="E60">A16</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_2_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_2"><i><font color="Green" title="E60">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_1"/><i><font color="Green" title="E61">A17</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2_1_1"><i><font color="Green" title="E61">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_2_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="measure5.html#T26" target="_self" title="MEASURE5:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_2_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="measure5.html#T28" target="_self" title="MEASURE5:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</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="E3:61_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_2"/><i><font color="Green" title="E61">A18</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2_1_2"><i><font color="Green" title="E61">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_2_1_1_1"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_2_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="measure5.html#T27" target="_self" title="MEASURE5:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_2_1_1_2"/>
<font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_2_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="measure5.html#T29" target="_self" title="MEASURE5:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_2"><i><font color="Green" title="E59">A15</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="E3:61_1_2_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:61_1_2"/><i><font color="Green" title="E59">A19</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b1</font> or <font color="Olive" title="b2">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Olive" title="b2">a2</font>,<font color="Olive" title="b4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:61_1_2_3"/><i><font color="Green" title="E60">A20</font></i>: 
( ( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_3"/><i><font color="Green" title="E61">A21</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_3_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_3"><i><font color="Green" title="E61">A21</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_1_1"/><i><font color="Green" title="E62">A22</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_3_1_1_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3_1_1"><i><font color="Green" title="E62">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_1_1_1_1_1"/>
<font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</font></i></a>, <a class="ref" href="measure5.html#T30" target="_self" title="MEASURE5:th.30">Th30</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_1_1_1_1_2"/>
<font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</font></i></a>, <a class="ref" href="measure5.html#T32" target="_self" title="MEASURE5:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_3_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</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="E3:61_1_2_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_1_2"/><i><font color="Green" title="E62">A23</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_3_1_2_1_1"/>
( <font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3_1_2"><i><font color="Green" title="E62">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_1_2_1_1_1"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</font></i></a>, <a class="ref" href="measure5.html#T31" target="_self" title="MEASURE5:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_1_2_1_1_2"/>
<font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</font></i></a>, <a class="ref" href="measure5.html#T33" target="_self" title="MEASURE5:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_3"><i><font color="Green" title="E60">A20</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="E3:61_1_2_3_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:61_1_2"/><i><font color="Green" title="E60">A24</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> implies ( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_2_4"/><i><font color="Green" title="E61">A25</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_4"/><i><font color="Green" title="E62">A26</font></i>: 
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_4_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_4"><i><font color="Green" title="E62">A26</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_4_1_1"/><i><font color="Green" title="E63">A27</font></i>: 
<font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_4_1_1_1_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_4_1_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_4_1_1_1_1_1"/>
<font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_4_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T34" target="_self" title="MEASURE5:th.34">Th34</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_4_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_4"><i><font color="Green" title="E61">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_4_1_1_1_1_2"/>
<font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_4_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T34" target="_self" title="MEASURE5:th.34">Th34</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_4_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_4"><i><font color="Green" title="E61">A25</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="E3:61_1_2_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_4_1_2"/><i><font color="Green" title="E63">A28</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_4_1_2_1_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font> or <font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_4_1_2"><i><font color="Green" title="E63">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_4_1_2_1_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_4_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T35" target="_self" title="MEASURE5:th.35">Th35</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_4_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_4"><i><font color="Green" title="E61">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_4_1_2_1_1_2"/>
<font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_4_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T35" target="_self" title="MEASURE5:th.35">Th35</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_4_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_4"><i><font color="Green" title="E61">A25</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="E3:61_1_2_4_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:61_1_2"/><i><font color="Green" title="E61">A29</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b1</font> or <font color="Olive" title="b2">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Olive" title="b2">a2</font>,<font color="Olive" title="b4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:61_1_2_5"/><i><font color="Green" title="E62">A30</font></i>: 
( ( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_5"/><i><font color="Green" title="E63">A31</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_5_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_5"><i><font color="Green" title="E63">A31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_5_1_1"/><i><font color="Green" title="E64">A32</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_5_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_5_1_1_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5_1_1"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_5_1_1_1_1_1"/>
<font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_5_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="measure5.html#T36" target="_self" title="MEASURE5:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_5_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_5_1_1_1_1_2"/>
<font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_5_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="measure5.html#T38" target="_self" title="MEASURE5:th.38">Th38</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_5_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</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="E3:61_1_2_5_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_5_1_2"/><i><font color="Green" title="E64">A33</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_5_1_2_1_1"/>
( <font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5_1_2"><i><font color="Green" title="E64">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_5_1_2_1_1_1"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_5_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="measure5.html#T37" target="_self" title="MEASURE5:th.37">Th37</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_5_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_5_1_2_1_1_2"/>
<font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_5_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="measure5.html#T39" target="_self" title="MEASURE5:th.39">Th39</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_5_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_5"><i><font color="Green" title="E62">A30</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="E3:61_1_2_5_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:61_1_2"/><i><font color="Green" title="E62">A34</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b1</font> or <font color="Olive" title="b2">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Olive" title="b2">a2</font>,<font color="Olive" title="b4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:61_1_2_6"/><i><font color="Green" title="E63">A35</font></i>: 
( ( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_6"/><i><font color="Green" title="E64">A36</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_6_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_6"><i><font color="Green" title="E64">A36</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_6_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_6_1_1"/><i><font color="Green" title="E65">A37</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_6_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_6_1_1_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6_1_1"><i><font color="Green" title="E65">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_6_1_1_1_1_1"/>
<font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_6_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="measure5.html#T40" target="_self" title="MEASURE5:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_6_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_6_1_1_1_1_2"/>
<font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_6_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="measure5.html#T42" target="_self" title="MEASURE5:th.42">Th42</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_6_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</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="E3:61_1_2_6_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_6_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_6_1_2"/><i><font color="Green" title="E65">A38</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_6_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_6_1_2_1_1"/>
( <font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6_1_2"><i><font color="Green" title="E65">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_6_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_6_1_2_1_1_1"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_6_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="measure5.html#T41" target="_self" title="MEASURE5:th.41">Th41</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_6_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_6_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_6_1_2_1_1_2"/>
<font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_6_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="measure5.html#T43" target="_self" title="MEASURE5:th.43">Th43</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_6_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_6"><i><font color="Green" title="E63">A35</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="E3:61_1_2_6_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:61_1_2"/><i><font color="Green" title="E63">A39</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> implies ( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_2_7"/><i><font color="Green" title="E64">A40</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_7"/><i><font color="Green" title="E65">A41</font></i>: 
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_7_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_7"><i><font color="Green" title="E65">A41</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_7_1_1"/><i><font color="Green" title="E66">A42</font></i>: 
<font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_7_1_1_1_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_7_1_1"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_7_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T44" target="_self" title="MEASURE5:th.44">Th44</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_7_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_7"><i><font color="Green" title="E64">A40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_7_1_1_1_1_2"/>
<font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_7_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T44" target="_self" title="MEASURE5:th.44">Th44</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_7_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_7"><i><font color="Green" title="E64">A40</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="E3:61_1_2_7_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_7_1_2"/><i><font color="Green" title="E66">A43</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_7_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_7_1_2_1_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font> or <font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_7_1_2"><i><font color="Green" title="E66">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_7_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_7_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T45" target="_self" title="MEASURE5:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_7_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_7"><i><font color="Green" title="E64">A40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_7_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_7_1_2_1_1_2"/>
<font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_7_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T45" target="_self" title="MEASURE5:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_7_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_7"><i><font color="Green" title="E64">A40</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="E3:61_1_2_7_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E9:61_1_2"/><i><font color="Green" title="E64">A44</font></i>: 
for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st ( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">b1</font> or <font color="Olive" title="b2">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Olive" title="b2">a2</font>,<font color="Olive" title="b4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> holds <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:61_1_2_8"/><i><font color="Green" title="E65">A45</font></i>: 
( ( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> ) &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_8"/><i><font color="Green" title="E66">A46</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_8_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_8"><i><font color="Green" title="E66">A46</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_8_1_1"/><i><font color="Green" title="E67">A47</font></i>: 
<font color="Maroon" title="c6">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_8_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_8_1_1_1_1"/>
( <font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8_1_1"><i><font color="Green" title="E67">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_8_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_8_1_1_1_1_1"/>
<font color="Maroon" title="c6">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_8_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</font></i></a>, <a class="ref" href="measure5.html#T46" target="_self" title="MEASURE5:th.46">Th46</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_8_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_8_1_1_1_1_2"/>
<font color="Maroon" title="c7">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_8_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</font></i></a>, <a class="ref" href="measure5.html#T48" target="_self" title="MEASURE5:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_8_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</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="E3:61_1_2_8_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_8_1_2"/><i><font color="Green" title="E67">A48</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_8_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_8_1_2_1_1"/>
( <font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8_1_2"><i><font color="Green" title="E67">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_8_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_8_1_2_1_1_1"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_8_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</font></i></a>, <a class="ref" href="measure5.html#T47" target="_self" title="MEASURE5:th.47">Th47</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_8_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_8_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_8_1_2_1_1_2"/>
<font color="Maroon" title="c9">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_8_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c7">a2</font>,<font color="Maroon" title="c9">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</font></i></a>, <a class="ref" href="measure5.html#T49" target="_self" title="MEASURE5:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_8_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_8"><i><font color="Green" title="E65">A45</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="E3:61_1_2_8_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:61_1_2"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> implies ( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_2_9"/><i><font color="Green" title="E65">A49</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:61_1_2_9"/><i><font color="Green" title="E66">A50</font></i>: 
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_9_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E2:61_1_2_9"><i><font color="Green" title="E66">A50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_9_1_1"/><i><font color="Green" title="E67">A51</font></i>: 
<font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_9_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_9_1_1_1_1"/>
( <font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_9_1_1"><i><font color="Green" title="E67">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_9_1_1_1_1_1"/>
<font color="Maroon" title="c1">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_9_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T50" target="_self" title="MEASURE5:th.50">Th50</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_9_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_9"><i><font color="Green" title="E65">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_9_1_1_1_1_2"/>
<font color="Maroon" title="c3">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_9_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T50" target="_self" title="MEASURE5:th.50">Th50</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_9_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_9"><i><font color="Green" title="E65">A49</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="E3:61_1_2_9_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_9_1_2"/><i><font color="Green" title="E67">A52</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_9_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_2_9_1_2_1_1"/>
( <font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font> or <font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_9_1_2"><i><font color="Green" title="E67">A52</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_9_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_9_1_2_1_1_1"/>
<font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_9_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T51" target="_self" title="MEASURE5:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_9_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_9"><i><font color="Green" title="E65">A49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_9_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_9_1_2_1_1_2"/>
<font color="Maroon" title="c4">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_9_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="measure5.html#T51" target="_self" title="MEASURE5:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_9_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:61_1_2_9"><i><font color="Green" title="E65">A49</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="E3:61_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="E11:61_1_2"/>
( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> &amp; <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:61"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="measure5.html#E2:61"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="measure5.html#E3:61"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="measure5.html#E1:61_1_2"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="measure5.html#E2:61_1_2"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="measure5.html#E3:61_1_2"><i><font color="Green" title="E58">A14</font></i></a>, <a class="txt" href="measure5.html#E4:61_1_2"><i><font color="Green" title="E59">A19</font></i></a>, <a class="txt" href="measure5.html#E5:61_1_2"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="measure5.html#E6:61_1_2"><i><font color="Green" title="E61">A29</font></i></a>, <a class="txt" href="measure5.html#E7:61_1_2"><i><font color="Green" title="E62">A34</font></i></a>, <a class="txt" href="measure5.html#E8:61_1_2"><i><font color="Green" title="E63">A39</font></i></a>, <a class="txt" href="measure5.html#E9:61_1_2"><i><font color="Green" title="E64">A44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
