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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>;<br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<a NAME="E2:69"/>
( <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c1">A</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 
<b>by </b><i><a class="ref" href="measure5.html#D9" target="_self" title="MEASURE5:def.9">Def9</a></i>;<br/>
<a NAME="E3:69"/><b>then </b>
(  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 
<b>by </b><i><a class="ref" href="measure5.html#D5" target="_self" title="MEASURE5:def.5">Def5</a>, <a class="ref" href="measure5.html#D6" target="_self" title="MEASURE5:def.6">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self" title="MEASURE5:def.7">Def7</a>, <a class="ref" href="measure5.html#D8" target="_self" title="MEASURE5:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">b1</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E5:69"/><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> or <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>
<a NAME="E6:69"/>
( <font color="Maroon" title="c2">B</font> is <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> or <font color="Maroon" title="c2">B</font> is <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a> or <font color="Maroon" title="c2">B</font> is <a href="measure5.html#V3" title="MEASURE5:attr.3">right_open_interval</a> or <font color="Maroon" title="c2">B</font> is <a href="measure5.html#V4" title="MEASURE5:attr.4">left_open_interval</a> )
 
<b>by </b><i><a class="ref" href="measure5.html#D9" target="_self" title="MEASURE5:def.9">Def9</a></i>;<br/>
<a NAME="E7:69"/><b>then </b>
(  ex <font color="Olive" title="b1">a2</font>, <font color="Olive" title="b2">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b2</font> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Olive" title="b1">a2</font>,<font color="Olive" title="b2">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or  ex <font color="Olive" title="b1">a2</font>, <font color="Olive" title="b2">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b2</font> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Olive" title="b1">a2</font>,<font color="Olive" title="b2">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a2</font>, <font color="Olive" title="b2">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b2</font> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Olive" title="b1">a2</font>,<font color="Olive" title="b2">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or  ex <font color="Olive" title="b1">a2</font>, <font color="Olive" title="b2">b2</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> st <br/>( <font color="Olive" title="b1">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b2</font> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Olive" title="b1">a2</font>,<font color="Olive" title="b2">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 
<b>by </b><i><a class="ref" href="measure5.html#D5" target="_self" title="MEASURE5:def.5">Def5</a>, <a class="ref" href="measure5.html#D6" target="_self" title="MEASURE5:def.6">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self" title="MEASURE5:def.7">Def7</a>, <a class="ref" href="measure5.html#D8" target="_self" title="MEASURE5:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c6">b2</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E9:69"/><i><font color="Green" title="E59">A3</font></i>: 
( <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> or <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1"/>
( ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> ) or ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="measure5.html#E5:69"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="measure5.html#E9:69"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>suppose </b></a><a NAME="E1:69_1_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_1_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_1"/><i><font color="Green" title="E61">A5</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_1"/><i><font color="Green" title="E62">A6</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_1_1_1_1_1"/><i><font color="Green" title="E63">A7</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1_1_1_1_1"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E64">A8</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_1_1_1_1_1_1_1"><i><font color="Green" title="E64">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E64">A9</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_1_1_1_1_1_1_2"><i><font color="Green" title="E64">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_1_1_1_1"/><i><font color="Green" title="E63">A10</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1_1_1_1"><i><font color="Green" title="E61">A5</font></i></a></i>;<br/><a NAME="E4:69_1_1_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_1_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_1_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_1_1_1_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_1_1_1_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_3"/><i><font color="Green" title="E61">A11</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E62">A12</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_1_1_1_3"/><i><font color="Green" title="E63">A13</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_1_1_1_3"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_1_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1_1_1_3"><i><font color="Green" title="E61">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_1_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_1_1_1_3"><i><font color="Green" title="E62">A12</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_1_1_1_3"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_1_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_2"><b>suppose </b></a><a NAME="E1:69_1_2"/><i><font color="Green" title="E60">A14</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_1"/><i><font color="Green" title="E61">A15</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_2_1_1_1"/><i><font color="Green" title="E62">A16</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_2_1_1_1_1"/><i><font color="Green" title="E63">A17</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2_1_1_1_1"><i><font color="Green" title="E63">A17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E64">A18</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2_1_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_2_1_1_1_1_1_1"><i><font color="Green" title="E64">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E64">A19</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2_1_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_2_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_2_1_1_1_1_1_2"><i><font color="Green" title="E64">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_2_1_1_1"/><i><font color="Green" title="E63">A20</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2_1_1_1"><i><font color="Green" title="E61">A15</font></i></a></i>;<br/><a NAME="E4:69_1_2_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_2_1_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_2_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_2_1_1_1"><i><font color="Green" title="E62">A16</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_2_1_1_1"><i><font color="Green" title="E63">A20</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_2_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self" title="MEASURE5:th.61">Th61</a></i>;<br/><b>hence </b><a NAME="E3:69_1_2_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_3"/><i><font color="Green" title="E61">A21</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E62">A22</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_2_1_1_3"/><i><font color="Green" title="E63">A23</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_2_1_1_3"><i><font color="Green" title="E61">A21</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_2_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2_1_1_3"><i><font color="Green" title="E61">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_2_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_2_1_1_3"><i><font color="Green" title="E62">A22</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_2_1_1_3"><i><font color="Green" title="E63">A23</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_2_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_2"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_2_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_3"><b>suppose </b></a><a NAME="E1:69_1_3"/><i><font color="Green" title="E60">A24</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_3_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_3_1_1_1"/><i><font color="Green" title="E61">A25</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_3_1_1_1"/><i><font color="Green" title="E62">A26</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_3_1_1_1_1"/><i><font color="Green" title="E63">A27</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3_1_1_1_1"><i><font color="Green" title="E63">A27</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E64">A28</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3_1_1_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_3_1_1_1_1_1_1"><i><font color="Green" title="E64">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E64">A29</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3_1_1_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_3_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_3_1_1_1_1_1_2"><i><font color="Green" title="E64">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_3_1_1_1"/><i><font color="Green" title="E63">A30</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3_1_1_1"><i><font color="Green" title="E61">A25</font></i></a></i>;<br/><a NAME="E4:69_1_3_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_3_1_1_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_3_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_3_1_1_1"><i><font color="Green" title="E62">A26</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_3_1_1_1"><i><font color="Green" title="E63">A30</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_3_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_3_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_3_1_1_3"/><i><font color="Green" title="E61">A31</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_3_1_1_3"/><b>then </b><i><font color="Green" title="E62">A32</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_3_1_1_3"/><i><font color="Green" title="E63">A33</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_3_1_1_3"><i><font color="Green" title="E61">A31</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_3_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3_1_1_3"><i><font color="Green" title="E61">A31</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_3_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_3_1_1_3"><i><font color="Green" title="E62">A32</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_3_1_1_3"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_3_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_3_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_3"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_3_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_4"><b>suppose </b></a><a NAME="E1:69_1_4"/><i><font color="Green" title="E60">A34</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_4_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_4_1_1_1"/><i><font color="Green" title="E61">A35</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_4_1_1_1"/><i><font color="Green" title="E62">A36</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_4_1_1_1_1"/><i><font color="Green" title="E63">A37</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_4_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4_1_1_1_1"><i><font color="Green" title="E63">A37</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_4_1_1_1_1_1_1"/><i><font color="Green" title="E64">A38</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4_1_1_1"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_4_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_4_1_1_1_1_1_1"><i><font color="Green" title="E64">A38</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_4_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_4_1_1_1_1_1_2"/><i><font color="Green" title="E64">A39</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4_1_1_1"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_4_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_4_1_1_1_1_1_2"><i><font color="Green" title="E64">A39</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_4_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_4_1_1_1"/><i><font color="Green" title="E63">A40</font></i>: 
( (  not <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; (  not <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4_1_1_1"><i><font color="Green" title="E61">A35</font></i></a></i>;<br/><a NAME="E4:69_1_4_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_4_1_1_1"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_4_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_4_1_1_1"><i><font color="Green" title="E62">A36</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_4_1_1_1"><i><font color="Green" title="E63">A40</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_4_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_4_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_4_1_1_3"/><i><font color="Green" title="E61">A41</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_4_1_1_3"/><b>then </b><i><font color="Green" title="E62">A42</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_4_1_1_3"/><i><font color="Green" title="E63">A43</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_4_1_1_3"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_4_1_1_3"/>
( ( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or <font color="Maroon" title="c5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; ( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <font color="Maroon" title="c5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4_1_1_3"><i><font color="Green" title="E61">A41</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_4_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_4_1_1_3"><i><font color="Green" title="E62">A42</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_4_1_1_3"><i><font color="Green" title="E63">A43</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_4_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_4_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_4_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_4"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_4_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_5"><b>suppose </b></a><a NAME="E1:69_1_5"/><i><font color="Green" title="E60">A44</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_5"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_5_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_5_1_1_1"/><i><font color="Green" title="E61">A45</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_5_1_1_1"/><i><font color="Green" title="E62">A46</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_5_1_1_1_1"/><i><font color="Green" title="E63">A47</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_5_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5_1_1_1_1"><i><font color="Green" title="E63">A47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_5_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_5_1_1_1_1_1_1"/><i><font color="Green" title="E64">A48</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5_1_1_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_5_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_5_1_1_1_1_1_1"><i><font color="Green" title="E64">A48</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_5_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_5_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_5_1_1_1_1_1_2"/><i><font color="Green" title="E64">A49</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5_1_1_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_5_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_5_1_1_1_1_1_2"><i><font color="Green" title="E64">A49</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_5_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_5_1_1_1"/><i><font color="Green" title="E63">A50</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5_1_1_1"><i><font color="Green" title="E61">A45</font></i></a></i>;<br/><a NAME="E4:69_1_5_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_5_1_1_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_5_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_5_1_1_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_5_1_1_1"><i><font color="Green" title="E63">A50</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_5_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_5_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_5_1_1_3"/><i><font color="Green" title="E61">A51</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_5_1_1_3"/><b>then </b><i><font color="Green" title="E62">A52</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_5_1_1_3"/><i><font color="Green" title="E63">A53</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_5_1_1_3"><i><font color="Green" title="E61">A51</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_5_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5_1_1_3"><i><font color="Green" title="E61">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_5_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_5_1_1_3"><i><font color="Green" title="E62">A52</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_5_1_1_3"><i><font color="Green" title="E63">A53</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_5_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_5_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_5"><i><font color="Green" title="E60">A44</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_5_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_6"><b>suppose </b></a><a NAME="E1:69_1_6"/><i><font color="Green" title="E60">A54</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_6"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_6_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_6_1_1_1"/><i><font color="Green" title="E61">A55</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_6_1_1_1"/><i><font color="Green" title="E62">A56</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_6_1_1_1_1"/><i><font color="Green" title="E63">A57</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_6_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6_1_1_1_1"><i><font color="Green" title="E63">A57</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_6_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_6_1_1_1_1_1_1"/><i><font color="Green" title="E64">A58</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6_1_1_1"><i><font color="Green" title="E61">A55</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_6_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_6_1_1_1_1_1_1"><i><font color="Green" title="E64">A58</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_6_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_6_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_6_1_1_1_1_1_2"/><i><font color="Green" title="E64">A59</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6_1_1_1"><i><font color="Green" title="E61">A55</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_6_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_6_1_1_1_1_1_2"><i><font color="Green" title="E64">A59</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_6_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_6_1_1_1"/><i><font color="Green" title="E63">A60</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6_1_1_1"><i><font color="Green" title="E61">A55</font></i></a></i>;<br/><a NAME="E4:69_1_6_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_6_1_1_1"><i><font color="Green" title="E61">A55</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_6_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_6_1_1_1"><i><font color="Green" title="E62">A56</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_6_1_1_1"><i><font color="Green" title="E63">A60</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_6_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_6_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self" title="MEASURE5:th.61">Th61</a></i>;<br/><b>hence </b><a NAME="E3:69_1_6_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_6_1_1_3"/><i><font color="Green" title="E61">A61</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_6_1_1_3"/><b>then </b><i><font color="Green" title="E62">A62</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_6_1_1_3"/><i><font color="Green" title="E63">A63</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_6_1_1_3"><i><font color="Green" title="E61">A61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_6_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6_1_1_3"><i><font color="Green" title="E61">A61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_6_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_6_1_1_3"><i><font color="Green" title="E62">A62</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_6_1_1_3"><i><font color="Green" title="E63">A63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_6_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_6_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_6_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_6"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_6_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_7"><b>suppose </b></a><a NAME="E1:69_1_7"/><i><font color="Green" title="E60">A64</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_7"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_7_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_7_1_1_1"/><i><font color="Green" title="E61">A65</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_7_1_1_1"/><i><font color="Green" title="E62">A66</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_7_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_7_1_1_1_1"/><i><font color="Green" title="E63">A67</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_7_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7_1_1_1_1"><i><font color="Green" title="E63">A67</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_7_1_1_1_1_1_1"/><i><font color="Green" title="E64">A68</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7_1_1_1"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_7_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_7_1_1_1_1_1_1"><i><font color="Green" title="E64">A68</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_7_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_7_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_7_1_1_1_1_1_2"/><i><font color="Green" title="E64">A69</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7_1_1_1"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_7_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_7_1_1_1_1_1_2"><i><font color="Green" title="E64">A69</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_7_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_7_1_1_1"/><i><font color="Green" title="E63">A70</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7_1_1_1"><i><font color="Green" title="E61">A65</font></i></a></i>;<br/><a NAME="E4:69_1_7_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_7_1_1_1"><i><font color="Green" title="E61">A65</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_7_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_7_1_1_1"><i><font color="Green" title="E62">A66</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_7_1_1_1"><i><font color="Green" title="E63">A70</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_7_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_7_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_7_1_1_3"/><i><font color="Green" title="E61">A71</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_7_1_1_3"/><b>then </b><i><font color="Green" title="E62">A72</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_7_1_1_3"/><i><font color="Green" title="E63">A73</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_7_1_1_3"><i><font color="Green" title="E61">A71</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_7_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7_1_1_3"><i><font color="Green" title="E61">A71</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_7_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_7_1_1_3"><i><font color="Green" title="E62">A72</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_7_1_1_3"><i><font color="Green" title="E63">A73</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_7_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_7_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_7_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_7"><i><font color="Green" title="E60">A64</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_7_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_8"><b>suppose </b></a><a NAME="E1:69_1_8"/><i><font color="Green" title="E60">A74</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_8"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_8_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_8_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_8_1_1_1"/><i><font color="Green" title="E61">A75</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_8_1_1_1"/><i><font color="Green" title="E62">A76</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_8_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_8_1_1_1_1"/><i><font color="Green" title="E63">A77</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_8_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8_1_1_1_1"><i><font color="Green" title="E63">A77</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_8_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_8_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_8_1_1_1_1_1_1"/><i><font color="Green" title="E64">A78</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8_1_1_1"><i><font color="Green" title="E61">A75</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_8_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_8_1_1_1_1_1_1"><i><font color="Green" title="E64">A78</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_8_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_8_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_8_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_8_1_1_1_1_1_2"/><i><font color="Green" title="E64">A79</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8_1_1_1"><i><font color="Green" title="E61">A75</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_8_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_8_1_1_1_1_1_2"><i><font color="Green" title="E64">A79</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:69_1_8_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_8_1_1_1"/><i><font color="Green" title="E63">A80</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8_1_1_1"><i><font color="Green" title="E61">A75</font></i></a></i>;<br/><a NAME="E4:69_1_8_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_8_1_1_1"><i><font color="Green" title="E61">A75</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_8_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_8_1_1_1"><i><font color="Green" title="E62">A76</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_8_1_1_1"><i><font color="Green" title="E63">A80</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_8_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_8_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_8_1_1_3"/><i><font color="Green" title="E61">A81</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_8_1_1_3"/><b>then </b><i><font color="Green" title="E62">A82</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_8_1_1_3"/><i><font color="Green" title="E63">A83</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_8_1_1_3"><i><font color="Green" title="E61">A81</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_8_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8_1_1_3"><i><font color="Green" title="E61">A81</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_8_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_8_1_1_3"><i><font color="Green" title="E62">A82</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_8_1_1_3"><i><font color="Green" title="E63">A83</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_8_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_8_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_8"><i><font color="Green" title="E60">A74</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_8_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_9"><b>suppose </b></a><a NAME="E1:69_1_9"/><i><font color="Green" title="E60">A84</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_9"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_9_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_9_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_9_1_1_1"/><i><font color="Green" title="E61">A85</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_9_1_1_1"/><i><font color="Green" title="E62">A86</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_9_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_9_1_1_1_1"/><i><font color="Green" title="E63">A87</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_9_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9_1_1_1_1"><i><font color="Green" title="E63">A87</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_9_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_9_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_9_1_1_1_1_1_1"/><i><font color="Green" title="E64">A88</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9_1_1_1"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_9_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_9_1_1_1_1_1_1"><i><font color="Green" title="E64">A88</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_9_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_9_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_9_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_9_1_1_1_1_1_2"/><i><font color="Green" title="E64">A89</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9_1_1_1"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_9_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_9_1_1_1_1_1_2"><i><font color="Green" title="E64">A89</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_9_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_9_1_1_1"/><i><font color="Green" title="E63">A90</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9_1_1_1"><i><font color="Green" title="E61">A85</font></i></a></i>;<br/><a NAME="E4:69_1_9_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_9_1_1_1"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_9_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_9_1_1_1"><i><font color="Green" title="E62">A86</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_9_1_1_1"><i><font color="Green" title="E63">A90</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_9_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_9_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_9_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_9_1_1_3"/><i><font color="Green" title="E61">A91</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_9_1_1_3"/><b>then </b><i><font color="Green" title="E62">A92</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_9_1_1_3"/><i><font color="Green" title="E63">A93</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_9_1_1_3"><i><font color="Green" title="E61">A91</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_9_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9_1_1_3"><i><font color="Green" title="E61">A91</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_9_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_9_1_1_3"><i><font color="Green" title="E62">A92</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_9_1_1_3"><i><font color="Green" title="E63">A93</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_9_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_9_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_9"><i><font color="Green" title="E60">A84</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_9_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_10"><b>suppose </b></a><a NAME="E1:69_1_10"/><i><font color="Green" title="E60">A94</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_10"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_10_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_10_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_10_1_1_1"/><i><font color="Green" title="E61">A95</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_10_1_1_1"/><i><font color="Green" title="E62">A96</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_10_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_10_1_1_1_1"/><i><font color="Green" title="E63">A97</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_10_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10_1_1_1_1"><i><font color="Green" title="E63">A97</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_10_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_10_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_10_1_1_1_1_1_1"/><i><font color="Green" title="E64">A98</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10_1_1_1"><i><font color="Green" title="E61">A95</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_10_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_10_1_1_1_1_1_1"><i><font color="Green" title="E64">A98</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_10_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_10_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_10_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_10_1_1_1_1_1_2"/><i><font color="Green" title="E64">A99</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10_1_1_1"><i><font color="Green" title="E61">A95</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_10_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_10_1_1_1_1_1_2"><i><font color="Green" title="E64">A99</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_10_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_10_1_1_1"/><i><font color="Green" title="E63">A100</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10_1_1_1"><i><font color="Green" title="E61">A95</font></i></a></i>;<br/><a NAME="E4:69_1_10_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_10_1_1_1"><i><font color="Green" title="E61">A95</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_10_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_10_1_1_1"><i><font color="Green" title="E62">A96</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_10_1_1_1"><i><font color="Green" title="E63">A100</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_10_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_10_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self" title="MEASURE5:th.61">Th61</a></i>;<br/><b>hence </b><a NAME="E3:69_1_10_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_10_1_1_3"/><i><font color="Green" title="E61">A101</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_10_1_1_3"/><b>then </b><i><font color="Green" title="E62">A102</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_10_1_1_3"/><i><font color="Green" title="E63">A103</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_10_1_1_3"><i><font color="Green" title="E61">A101</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_10_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10_1_1_3"><i><font color="Green" title="E61">A101</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_10_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_10_1_1_3"><i><font color="Green" title="E62">A102</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_10_1_1_3"><i><font color="Green" title="E63">A103</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_10_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_10_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_10_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_10"><i><font color="Green" title="E60">A94</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_10_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_11"><b>suppose </b></a><a NAME="E1:69_1_11"/><i><font color="Green" title="E60">A104</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_11"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_11_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_11_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_11_1_1_1"/><i><font color="Green" title="E61">A105</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_11_1_1_1"/><i><font color="Green" title="E62">A106</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_11_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_11_1_1_1_1"/><i><font color="Green" title="E63">A107</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_11_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11_1_1_1_1"><i><font color="Green" title="E63">A107</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_11_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_11_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_11_1_1_1_1_1_1"/><i><font color="Green" title="E64">A108</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11_1_1_1"><i><font color="Green" title="E61">A105</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_11_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_11_1_1_1_1_1_1"><i><font color="Green" title="E64">A108</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_11_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_11_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_11_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_11_1_1_1_1_1_2"/><i><font color="Green" title="E64">A109</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11_1_1_1"><i><font color="Green" title="E61">A105</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_11_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_11_1_1_1_1_1_2"><i><font color="Green" title="E64">A109</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_11_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_11_1_1_1"/><i><font color="Green" title="E63">A110</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11_1_1_1"><i><font color="Green" title="E61">A105</font></i></a></i>;<br/><a NAME="E4:69_1_11_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_11_1_1_1"><i><font color="Green" title="E61">A105</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_11_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_11_1_1_1"><i><font color="Green" title="E62">A106</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_11_1_1_1"><i><font color="Green" title="E63">A110</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_11_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_11_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_11_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_11_1_1_3"/><i><font color="Green" title="E61">A111</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_11_1_1_3"/><b>then </b><i><font color="Green" title="E62">A112</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_11_1_1_3"/><i><font color="Green" title="E63">A113</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_11_1_1_3"><i><font color="Green" title="E61">A111</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_11_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11_1_1_3"><i><font color="Green" title="E61">A111</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_11_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_11_1_1_3"><i><font color="Green" title="E62">A112</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_11_1_1_3"><i><font color="Green" title="E63">A113</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_11_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_11_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_11_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_11"><i><font color="Green" title="E60">A104</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_11_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_12"><b>suppose </b></a><a NAME="E1:69_1_12"/><i><font color="Green" title="E60">A114</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_12"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_12_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_12_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_12_1_1_1"/><i><font color="Green" title="E61">A115</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_12_1_1_1"/><i><font color="Green" title="E62">A116</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_12_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_12_1_1_1_1"/><i><font color="Green" title="E63">A117</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_12_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12_1_1_1_1"><i><font color="Green" title="E63">A117</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_12_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_12_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_12_1_1_1_1_1_1"/><i><font color="Green" title="E64">A118</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12_1_1_1"><i><font color="Green" title="E61">A115</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_12_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_12_1_1_1_1_1_1"><i><font color="Green" title="E64">A118</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_12_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_12_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_12_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_12_1_1_1_1_1_2"/><i><font color="Green" title="E64">A119</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12_1_1_1"><i><font color="Green" title="E61">A115</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_12_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_12_1_1_1_1_1_2"><i><font color="Green" title="E64">A119</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:69_1_12_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_12_1_1_1"/><i><font color="Green" title="E63">A120</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12_1_1_1"><i><font color="Green" title="E61">A115</font></i></a></i>;<br/><a NAME="E4:69_1_12_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_12_1_1_1"><i><font color="Green" title="E61">A115</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_12_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_12_1_1_1"><i><font color="Green" title="E62">A116</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_12_1_1_1"><i><font color="Green" title="E63">A120</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_12_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_12_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_12_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_12_1_1_3"/><i><font color="Green" title="E61">A121</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_12_1_1_3"/><b>then </b><i><font color="Green" title="E62">A122</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_12_1_1_3"/><i><font color="Green" title="E63">A123</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_12_1_1_3"><i><font color="Green" title="E61">A121</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_12_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12_1_1_3"><i><font color="Green" title="E61">A121</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_12_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_12_1_1_3"><i><font color="Green" title="E62">A122</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_12_1_1_3"><i><font color="Green" title="E63">A123</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_12_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_12_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_12"><i><font color="Green" title="E60">A114</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_12_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_13"><b>suppose </b></a><a NAME="E1:69_1_13"/><i><font color="Green" title="E60">A124</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K2" title="MEASURE5:func.2">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_13"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_13_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_13_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_13_1_1_1"/><i><font color="Green" title="E61">A125</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_13_1_1_1"/><i><font color="Green" title="E62">A126</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_13_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_13_1_1_1_1"/><i><font color="Green" title="E63">A127</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_13_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13_1_1_1_1"><i><font color="Green" title="E63">A127</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_13_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_13_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_13_1_1_1_1_1_1"/><i><font color="Green" title="E64">A128</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13_1_1_1"><i><font color="Green" title="E61">A125</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_13_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_13_1_1_1_1_1_1"><i><font color="Green" title="E64">A128</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_13_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_13_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_13_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_13_1_1_1_1_1_2"/><i><font color="Green" title="E64">A129</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13_1_1_1"><i><font color="Green" title="E61">A125</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_13_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_13_1_1_1_1_1_2"><i><font color="Green" title="E64">A129</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_13_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self" title="MEASURE5:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_13_1_1_1"/><i><font color="Green" title="E63">A130</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13_1_1_1"><i><font color="Green" title="E61">A125</font></i></a></i>;<br/><a NAME="E4:69_1_13_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_13_1_1_1"><i><font color="Green" title="E61">A125</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_13_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_13_1_1_1"><i><font color="Green" title="E62">A126</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_13_1_1_1"><i><font color="Green" title="E63">A130</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_13_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_13_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_13_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_13_1_1_3"/><i><font color="Green" title="E61">A131</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_13_1_1_3"/><b>then </b><i><font color="Green" title="E62">A132</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_13_1_1_3"/><i><font color="Green" title="E63">A133</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_13_1_1_3"><i><font color="Green" title="E61">A131</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_13_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13_1_1_3"><i><font color="Green" title="E61">A131</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_13_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_13_1_1_3"><i><font color="Green" title="E62">A132</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_13_1_1_3"><i><font color="Green" title="E63">A133</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_13_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_13_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_13_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_13"><i><font color="Green" title="E60">A124</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_13_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_14"><b>suppose </b></a><a NAME="E1:69_1_14"/><i><font color="Green" title="E60">A134</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_14"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_14_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_14_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_14_1_1_1"/><i><font color="Green" title="E61">A135</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_14_1_1_1"/><i><font color="Green" title="E62">A136</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_14_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_14_1_1_1_1"/><i><font color="Green" title="E63">A137</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_14_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14_1_1_1_1"><i><font color="Green" title="E63">A137</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_14_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_14_1_1_1_1_1_1"/><i><font color="Green" title="E64">A138</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14_1_1_1"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_14_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_14_1_1_1_1_1_1"><i><font color="Green" title="E64">A138</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_14_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_14_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_14_1_1_1_1_1_2"/><i><font color="Green" title="E64">A139</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14_1_1_1"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_14_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_14_1_1_1_1_1_2"><i><font color="Green" title="E64">A139</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_14_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self" title="MEASURE5:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_14_1_1_1"/><i><font color="Green" title="E63">A140</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14_1_1_1"><i><font color="Green" title="E61">A135</font></i></a></i>;<br/><a NAME="E4:69_1_14_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_14_1_1_1"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_14_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_14_1_1_1"><i><font color="Green" title="E62">A136</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_14_1_1_1"><i><font color="Green" title="E63">A140</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_14_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_14_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self" title="MEASURE5:th.61">Th61</a></i>;<br/><b>hence </b><a NAME="E3:69_1_14_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_14_1_1_3"/><i><font color="Green" title="E61">A141</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_14_1_1_3"/><b>then </b><i><font color="Green" title="E62">A142</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_14_1_1_3"/><i><font color="Green" title="E63">A143</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_14_1_1_3"><i><font color="Green" title="E61">A141</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_14_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14_1_1_3"><i><font color="Green" title="E61">A141</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_14_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_14_1_1_3"><i><font color="Green" title="E62">A142</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_14_1_1_3"><i><font color="Green" title="E63">A143</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_14_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_14_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_14"><i><font color="Green" title="E60">A134</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_14_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_15"><b>suppose </b></a><a NAME="E1:69_1_15"/><i><font color="Green" title="E60">A144</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_15"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_15_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_15_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_15_1_1_1"/><i><font color="Green" title="E61">A145</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_15_1_1_1"/><i><font color="Green" title="E62">A146</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_15_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_15_1_1_1_1"/><i><font color="Green" title="E63">A147</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_15_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15_1_1_1_1"><i><font color="Green" title="E63">A147</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_15_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_15_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_15_1_1_1_1_1_1"/><i><font color="Green" title="E64">A148</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15_1_1_1"><i><font color="Green" title="E61">A145</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_15_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_15_1_1_1_1_1_1"><i><font color="Green" title="E64">A148</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_15_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_15_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_15_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_15_1_1_1_1_1_2"/><i><font color="Green" title="E64">A149</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15_1_1_1"><i><font color="Green" title="E61">A145</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_15_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_15_1_1_1_1_1_2"><i><font color="Green" title="E64">A149</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_15_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_15_1_1_1"/><i><font color="Green" title="E63">A150</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15_1_1_1"><i><font color="Green" title="E61">A145</font></i></a></i>;<br/><a NAME="E4:69_1_15_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_15_1_1_1"><i><font color="Green" title="E61">A145</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_15_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_15_1_1_1"><i><font color="Green" title="E62">A146</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_15_1_1_1"><i><font color="Green" title="E63">A150</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_15_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_15_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_15_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_15_1_1_3"/><i><font color="Green" title="E61">A151</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_15_1_1_3"/><b>then </b><i><font color="Green" title="E62">A152</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_15_1_1_3"/><i><font color="Green" title="E63">A153</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_15_1_1_3"><i><font color="Green" title="E61">A151</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_15_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15_1_1_3"><i><font color="Green" title="E61">A151</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_15_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_15_1_1_3"><i><font color="Green" title="E62">A152</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_15_1_1_3"><i><font color="Green" title="E63">A153</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_15_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_15_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_15"><i><font color="Green" title="E60">A144</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_15_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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="69_1_16"><b>suppose </b></a><a NAME="E1:69_1_16"/><i><font color="Green" title="E60">A154</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K3" title="MEASURE5:func.3">].</a><span class="default"><font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:69_1_16"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_16_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_16_1_1"/>
( ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> ) or ( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_16_1_1_1"/><i><font color="Green" title="E61">A155</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_16_1_1_1"/><i><font color="Green" title="E62">A156</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_16_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_16_1_1_1_1"/><i><font color="Green" title="E63">A157</font></i>: 
(  not <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> or  not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_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="E1:69_1_16_1_1_1_1_1"/>
( <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16_1_1_1_1"><i><font color="Green" title="E63">A157</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_16_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_16_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_16_1_1_1_1_1_1"/><i><font color="Green" title="E64">A158</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16_1_1_1"><i><font color="Green" title="E61">A155</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self" title="MEASURE5:th.19">Th19</a></i>;<br/><a NAME="E4:69_1_16_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_16_1_1_1_1_1_1"><i><font color="Green" title="E64">A158</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_16_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_16_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_16_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</font> being   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a><b> such that </b><br/><a NAME="E3:69_1_16_1_1_1_1_1_2"/><i><font color="Green" title="E64">A159</font></i>: 
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16_1_1_1"><i><font color="Green" title="E61">A155</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self" title="MEASURE5:th.18">Th18</a></i>;<br/><a NAME="E4:69_1_16_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K4" title="MEASURE5:func.4">[.</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; not <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:69_1_16_1_1_1_1_1_2"><i><font color="Green" title="E64">A159</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self" title="MEASURE5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:69_1_16_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self" title="MEASURE5:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:69_1_16_1_1_1"/><i><font color="Green" title="E63">A160</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16_1_1_1"><i><font color="Green" title="E61">A155</font></i></a></i>;<br/><a NAME="E4:69_1_16_1_1_1"/>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">a1</font> &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_16_1_1_1"><i><font color="Green" title="E61">A155</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:69_1_16_1_1_1"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_16_1_1_1"><i><font color="Green" title="E62">A156</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_16_1_1_1"><i><font color="Green" title="E63">A160</font></i></a>, <a class="ref" href="supinf_2.html#T15" title="SUPINF_2:th.15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_16_1_1_2"/>
( <font color="Maroon" title="c3">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_16_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self" title="MEASURE5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:69_1_16_1_1_2"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:69_1_16_1_1_3"/><i><font color="Green" title="E61">A161</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_16_1_1_3"/><b>then </b><i><font color="Green" title="E62">A162</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E3:69_1_16_1_1_3"/><i><font color="Green" title="E63">A163</font></i>: 
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b2</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="txt" href="measure5.html#E1:69_1_16_1_1_3"><i><font color="Green" title="E61">A161</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:69_1_16_1_1_3"/>
( (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; (  not <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or  not <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) &amp; <font color="Maroon" title="c5">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16_1_1_3"><i><font color="Green" title="E61">A161</font></i></a></i>;<br/><b>hence </b><a NAME="E5:69_1_16_1_1_3"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:69_1_16_1_1_3"><i><font color="Green" title="E62">A162</font></i></a>, <a class="txt" href="measure5.html#E3:69_1_16_1_1_3"><i><font color="Green" title="E63">A163</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self" title="MEASURE5:th.2">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_16_1_1_4"><b>suppose </b></a><a NAME="E1:69_1_16_1_1_4"/>
( <font color="Maroon" title="c4">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font> )
 ;<br/><div class="add"><a NAME="E2:69_1_16_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp;  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:69_1_16"><i><font color="Green" title="E60">A154</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:69_1_16_1_1_4"/>
 <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure5.html#K5" title="MEASURE5:func.5">vol</a> <font color="Maroon" title="c2">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>
