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<div class="add">

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



<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c2">a</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">b</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>  &amp; <font color="Maroon" title="c4">c</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="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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</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="c2">a</font>,<font color="Maroon" title="c3">b</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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_1"/><b>then </b>
( <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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E3:63_1_1"/><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> <font color="Maroon" title="c3">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:63_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>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="supinf_1.html#T6" title="SUPINF_1:th.6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T7" title="SUPINF_2:th.7">SUPINF_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/>
<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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_2"/><b>then </b>
( <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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K2" title="MEASURE5:func.2">.[</a></span> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><a NAME="E3:63_1_2"/><b>then </b><i><font color="Green" title="E55">A2</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> <font color="Maroon" title="c4">c</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:63_1_2"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">c</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">b</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">c</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">b</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#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D2" title="XXREAL_0:def.2">XXREAL_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:63_1_2"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="measure5.html#E3:63_1_2"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">SUPINF_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_3"><b>suppose </b></a><a NAME="E1:63_1_3"/><i><font color="Green" title="E55">A3</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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_3"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E3:63_1_3"/><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> <font color="Maroon" title="c3">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:63_1_3"><i><font color="Green" title="E55">A3</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="E4:63_1_3"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="supinf_1.html#T6" title="SUPINF_1:th.6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T7" title="SUPINF_2:th.7">SUPINF_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_4"><b>suppose </b></a><a NAME="E1:63_1_4"/><i><font color="Green" title="E55">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#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_4"/>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><a NAME="E3:63_1_4"/><b>then </b><i><font color="Green" title="E56">A5</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> <font color="Maroon" title="c4">c</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:63_1_4"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:63_1_4"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">c</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">b</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">c</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">b</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#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D2" title="XXREAL_0:def.2">XXREAL_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:63_1_4"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="measure5.html#E3:63_1_4"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">SUPINF_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_5"><b>suppose </b></a><a NAME="E1:63_1_5"/><i><font color="Green" title="E55">A6</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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_5"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E3:63_1_5"/><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> <font color="Maroon" title="c3">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:63_1_5"><i><font color="Green" title="E55">A6</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="E4:63_1_5"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="supinf_1.html#T6" title="SUPINF_1:th.6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T7" title="SUPINF_2:th.7">SUPINF_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_6"><b>suppose </b></a><a NAME="E1:63_1_6"/>
<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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_6"/><b>then </b>
( <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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K4" title="MEASURE5:func.4">.[</a></span> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><a NAME="E3:63_1_6"/><b>then </b><i><font color="Green" title="E55">A7</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> <font color="Maroon" title="c4">c</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:63_1_6"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">c</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">b</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">c</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">b</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#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D2" title="XXREAL_0:def.2">XXREAL_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:63_1_6"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="measure5.html#E3:63_1_6"><i><font color="Green" title="E55">A7</font></i></a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">SUPINF_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_7"><b>suppose </b></a><a NAME="E1:63_1_7"/>
<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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_7"/><b>then </b>
( <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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><a NAME="E3:63_1_7"/><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> <font color="Maroon" title="c3">b</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:63_1_7"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="supinf_1.html#T6" title="SUPINF_1:th.6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#T7" title="SUPINF_2:th.7">SUPINF_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_8"><b>suppose </b></a><a NAME="E1:63_1_8"/>
<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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:63_1_8"/><b>then </b>
( <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">b</font>,<font color="Maroon" title="c4">c</font></span><a href="measure5.html#K3" title="MEASURE5:func.3">.]</a></span> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/><a NAME="E3:63_1_8"/><b>then </b><i><font color="Green" title="E55">A8</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> <font color="Maroon" title="c4">c</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="measure5.html#D10" target="_self" title="MEASURE5:def.10">Def10</a></i>;<br/><a NAME="E4:63_1_8"/>
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c4">c</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">b</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">c</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">b</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#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="xxreal_0.html#D2" title="XXREAL_0:def.2">XXREAL_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:63_1_8"/>
 <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_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:63"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="measure5.html#E3:63_1_8"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="supinf_2.html#T6" title="SUPINF_2:th.6">SUPINF_2:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
