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

<b>let </b><font color="Maroon">A</font> be   <a href="measure5.html#NM1">Interval</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="measure5.html#NM1">Interval</a>;<br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:67"/>
( <font color="Maroon">A</font> is <a href="measure5.html#V1" target="_self">open_interval</a> or <font color="Maroon">A</font> is <a href="measure5.html#V2" target="_self">closed_interval</a> or <font color="Maroon">A</font> is <a href="measure5.html#V3" target="_self">right_open_interval</a> or <font color="Maroon">A</font> is <a href="measure5.html#V4" target="_self">left_open_interval</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:67"/><b>then </b>
(  ex <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or  ex <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or  ex <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or  ex <font color="Olive">a1</font>, <font color="Olive">b1</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:67"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>
<a NAME="E6:67"/>
( <font color="Maroon">B</font> is <a href="measure5.html#V1" target="_self">open_interval</a> or <font color="Maroon">B</font> is <a href="measure5.html#V2" target="_self">closed_interval</a> or <font color="Maroon">B</font> is <a href="measure5.html#V3" target="_self">right_open_interval</a> or <font color="Maroon">B</font> is <a href="measure5.html#V4" target="_self">left_open_interval</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:67"/><b>then </b>
(  ex <font color="Olive">a2</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b2</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or  ex <font color="Olive">a2</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b2</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or  ex <font color="Olive">a2</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b2</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or  ex <font color="Olive">a2</font>, <font color="Olive">b2</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b2</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">a2</font>,<font color="Olive">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">a2</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b2</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E9:67"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 </span><b>by </b><i><a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_1_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_1_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_1_1_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_1_1_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_1_1_1_1"/><i><font color="Green">E35</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_1_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_1_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_3"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_1_1_1_3"/><i><font color="Green">E38</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_1_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_1_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_2_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_2_1_1_1_1"/><i><font color="Green">E42</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1_2_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_2_1_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_2_1_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_2_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_2_1_1_1"/><i><font color="Green">E45</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_2_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:67_1_2_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_3"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_2_1_1_3"/><i><font color="Green">E48</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_2_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_2_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_3"><b>suppose </b></a><a NAME="E1:67_1_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_3_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_3_1_1_1_1"/><i><font color="Green">E112</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_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="E1:67_1_3_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_3_1_1_1_1_1_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_3_1_1_1_1_1_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_3_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_3_1_1_1"/><i><font color="Green">E115</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_3_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_3_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_3"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_3"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_3_1_1_3"/><i><font color="Green">E118</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_3_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_3_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_4"><b>suppose </b></a><a NAME="E1:67_1_4"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_4_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_4_1_1_1_1"/><i><font color="Green">E122</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_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="E1:67_1_4_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_4_1_1_1_1_1_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_4_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_4_1_1_1_1_1_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_4_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_4_1_1_1"/><i><font color="Green">E125</font></i>: 
( (  not <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; (  not <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_4_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_4_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_3"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_3"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_4_1_1_3"/><i><font color="Green">E128</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_4_1_1_3"/>
( ( <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; ( <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_4_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_5"><b>suppose </b></a><a NAME="E1:67_1_5"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_5"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_5_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_1"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_5_1_1_1_1"/><i><font color="Green">E132</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_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="E1:67_1_5_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_5_1_1_1_1_1_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_5_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_5_1_1_1_1_1_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_5_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_5_1_1_1"/><i><font color="Green">E135</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_5_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_5_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_3"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_3"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_5_1_1_3"/><i><font color="Green">E145</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_5_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_5_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_6"><b>suppose </b></a><a NAME="E1:67_1_6"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_6"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_6_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_6_1_1_1_1"/><i><font color="Green">E149</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_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="E1:67_1_6_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_6_1_1_1_1_1_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_6_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_6_1_1_1_1_1_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_6_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_6_1_1_1"/><i><font color="Green">E152</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_6_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:67_1_6_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_3"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_3"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_6_1_1_3"/><i><font color="Green">E155</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_6_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_6_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_7"><b>suppose </b></a><a NAME="E1:67_1_7"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_7"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_7_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_1"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_7_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_7_1_1_1_1"/><i><font color="Green">E159</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_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="E1:67_1_7_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_7_1_1_1_1_1_1"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_7_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_7_1_1_1_1_1_2"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_7_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_7_1_1_1"/><i><font color="Green">E162</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_7_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_7_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_3"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_3"/><b>then </b><i><font color="Green">E164</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_7_1_1_3"/><i><font color="Green">E165</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_7_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_7_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_8"><b>suppose </b></a><a NAME="E1:67_1_8"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_8"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_8_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_8_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_1"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_8_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_8_1_1_1_1"/><i><font color="Green">E169</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_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="E1:67_1_8_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_8_1_1_1_1_1_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_8_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_8_1_1_1_1_1_2"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_8_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_8_1_1_1"/><i><font color="Green">E172</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_8_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_8_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_3"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_3"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_8_1_1_3"/><i><font color="Green">E175</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_8_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_8_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_9"><b>suppose </b></a><a NAME="E1:67_1_9"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_9"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_9_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_9_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_1"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_1"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_9_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_9_1_1_1_1"/><i><font color="Green">E179</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_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="E1:67_1_9_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_9_1_1_1_1_1_1"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_9_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_9_1_1_1_1_1_2"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_9_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_9_1_1_1"/><i><font color="Green">E182</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_9_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_9_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_3"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_3"/><b>then </b><i><font color="Green">E184</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_9_1_1_3"/><i><font color="Green">E185</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_9_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_9_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_10"><b>suppose </b></a><a NAME="E1:67_1_10"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_10"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_10_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_10_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_1"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_1"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_10_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_10_1_1_1_1"/><i><font color="Green">E189</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_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="E1:67_1_10_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_10_1_1_1_1_1_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_10_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_10_1_1_1_1_1_2"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_10_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_10_1_1_1"/><i><font color="Green">E192</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_10_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:67_1_10_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_3"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_3"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_10_1_1_3"/><i><font color="Green">E195</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_10_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_10_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_11"><b>suppose </b></a><a NAME="E1:67_1_11"/><i><font color="Green">E196</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_11"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_11_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_11_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_1"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_1"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_11_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_11_1_1_1_1"/><i><font color="Green">E199</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_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="E1:67_1_11_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_11_1_1_1_1_1_1"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_11_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_11_1_1_1_1_1_2"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_11_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_11_1_1_1"/><i><font color="Green">E202</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_11_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_11_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_3"/><i><font color="Green">E203</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_3"/><b>then </b><i><font color="Green">E204</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_11_1_1_3"/><i><font color="Green">E205</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_11_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_11_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_12"><b>suppose </b></a><a NAME="E1:67_1_12"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_12"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_12_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_12_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_1"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_1"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_12_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_12_1_1_1_1"/><i><font color="Green">E209</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_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="E1:67_1_12_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_12_1_1_1_1_1_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_12_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_12_1_1_1_1_1_2"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_12_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_12_1_1_1"/><i><font color="Green">E212</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_12_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_12_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_3"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_3"/><b>then </b><i><font color="Green">E214</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_12_1_1_3"/><i><font color="Green">E215</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_12_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_12_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_13"><b>suppose </b></a><a NAME="E1:67_1_13"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_13"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_13_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_13_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_1"/><i><font color="Green">E217</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_1"/><i><font color="Green">E218</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_13_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_13_1_1_1_1"/><i><font color="Green">E219</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_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="E1:67_1_13_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_13_1_1_1_1_1_1"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_13_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_13_1_1_1_1_1_2"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_13_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_13_1_1_1"/><i><font color="Green">E222</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_13_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_13_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_3"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_3"/><b>then </b><i><font color="Green">E224</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_13_1_1_3"/><i><font color="Green">E225</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_13_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_13_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_14"><b>suppose </b></a><a NAME="E1:67_1_14"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_14"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_14_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_14_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_1"/><i><font color="Green">E227</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_1"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_14_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_14_1_1_1_1"/><i><font color="Green">E229</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_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="E1:67_1_14_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_14_1_1_1_1_1_1"/><i><font color="Green">E230</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_14_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_14_1_1_1_1_1_2"/><i><font color="Green">E231</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_14_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_14_1_1_1"/><i><font color="Green">E232</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_14_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:67_1_14_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_3"/><i><font color="Green">E233</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_3"/><b>then </b><i><font color="Green">E234</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_14_1_1_3"/><i><font color="Green">E235</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_14_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_14_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_15"><b>suppose </b></a><a NAME="E1:67_1_15"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_15"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_15_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_15_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_1"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_1"/><i><font color="Green">E238</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_15_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_15_1_1_1_1"/><i><font color="Green">E239</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_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="E1:67_1_15_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_15_1_1_1_1_1_1"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_15_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_15_1_1_1_1_1_2"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_15_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_15_1_1_1"/><i><font color="Green">E242</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_15_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_15_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_3"/><i><font color="Green">E243</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_3"/><b>then </b><i><font color="Green">E244</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_15_1_1_3"/><i><font color="Green">E245</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_15_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_15_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<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="67_1_16"><b>suppose </b></a><a NAME="E1:67_1_16"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_16"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_16_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_16_1_1"/>
( ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> ) or ( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_1"/><i><font color="Green">E247</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_1"/><i><font color="Green">E248</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_16_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_16_1_1_1_1"/><i><font color="Green">E249</font></i>: 
(  not <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> or  not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_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="E1:67_1_16_1_1_1_1_1"/>
( <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> or <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> )
 </span><b>by </b><i><a class="ref" href="measure5.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_16_1_1_1_1_1_1"/><i><font color="Green">E250</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_16_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_1_1_1_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_16_1_1_1_1_1_2"/><i><font color="Green">E251</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_16_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_16_1_1_1"/><i><font color="Green">E252</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:67_1_16_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a1</font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_2"/>
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_16_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_3"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_3"/><b>then </b><i><font color="Green">E254</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:67_1_16_1_1_3"/><i><font color="Green">E255</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="measure5.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E4:67_1_16_1_1_3"/>
( (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_4"/>
( <font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:67_1_16_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
