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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="measure5.html#NM1">Interval</a>;<br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:67"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V1" target="_self">open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V2" target="_self">closed_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V3" target="_self">right_open_interval</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="measure5.html#V4" target="_self">left_open_interval</a> )
 
<b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:67"/><b>then </b>
( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 
<b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, <a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:67"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>
<a NAME="E6:67"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="measure5.html#V1" target="_self">open_interval</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="measure5.html#V2" target="_self">closed_interval</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="measure5.html#V3" target="_self">right_open_interval</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="measure5.html#V4" target="_self">left_open_interval</a> )
 
<b>by </b><i><a class="ref" href="measure5.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E7:67"/><b>then </b>
( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 
<b>by </b><i><a class="ref" href="measure5.html#D5" target="_self">Def5</a>, <a class="ref" href="measure5.html#D6" target="_self">Def6</a>, <a class="ref" href="measure5.html#D7" target="_self">Def7</a>, <a class="ref" href="measure5.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E9:67"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> ) )
 </span><b>by </b><i><a class="txt" href="measure5.html#E5:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="measure5.html#E9:67"><i><font color="Green">E59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_1_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_1_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_1_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_1_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_1_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_1_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_1_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_1_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_1_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_1_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_1_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_1_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_1_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_1_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_1_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_1_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_1_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_2_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_2_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_2_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_2_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_2_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_2_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_2_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_2_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_2_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_2_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_2_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_2_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self">Th61</a></i>;<br/><b>hence </b><a NAME="E3:67_1_2_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_2_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_2_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_2_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_2_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_2_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_2_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_2_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_2_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_2_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3"><b>suppose </b></a><a NAME="E1:67_1_3"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_3_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_3_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_3_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_3_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_3_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_3_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_3_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_3_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_3_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_3_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_3_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_3_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_3_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_3_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_3_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_3_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_3_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_3_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_3_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_3_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4"><b>suppose </b></a><a NAME="E1:67_1_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_4_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_4_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_4_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_4_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_4_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_4_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_4_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_4_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_4_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_4_1_1_1"/><i><font color="Green">E63</font></i>: 
( (  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; (  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_4_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_4_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_4_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_4_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_4_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_4_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_4_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_4_1_1_3"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_4_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_4_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_4_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_4_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_4_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_4_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5"><b>suppose </b></a><a NAME="E1:67_1_5"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_5"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_5_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_5_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_5_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_5_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_5_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_5_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_5_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_5_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_5_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_5_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_5_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_5_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_5_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_5_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_5_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_5_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_5_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_5_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_5_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_5_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_5_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_5_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_5_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_5_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6"><b>suppose </b></a><a NAME="E1:67_1_6"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_6"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_6_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_6_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_6_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_6_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_6_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_6_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_6_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_6_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_6_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_6_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_6_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_6_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_6_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_6_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self">Th61</a></i>;<br/><b>hence </b><a NAME="E3:67_1_6_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_6_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_6_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_6_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_6_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_6_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_6_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_6_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_6_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_6_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_6_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7"><b>suppose </b></a><a NAME="E1:67_1_7"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_7"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_7_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_7_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_7_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_7_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_7_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_7_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_7_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_7_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_7_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_7_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_7_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_7_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_7_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_7_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_7_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_7_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_7_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_7_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_7_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_7_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_7_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_7_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_7_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_7_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_7_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_7"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_7_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8"><b>suppose </b></a><a NAME="E1:67_1_8"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_8"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_8_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_8_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_8_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_8_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_8_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_8_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_8_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_8_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_8_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_8_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_8_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_8_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_8_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_8_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_8_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_8_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_8_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_8_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_8_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_8_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_8_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_8_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_8_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_8_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_8_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_8"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_8_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9"><b>suppose </b></a><a NAME="E1:67_1_9"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_9"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_9_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_9_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_9_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_9_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_9_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_9_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_9_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_9_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_9_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_9_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_9_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_9_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_9_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_9_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_9_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_9_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_9_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_9_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_9_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_9_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_9_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_9_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_9_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_9_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_9_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_9"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_9_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10"><b>suppose </b></a><a NAME="E1:67_1_10"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_10"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_10_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_10_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_10_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_10_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_10_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_10_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_10_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_10_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_10_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_10_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_10_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_10_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_10_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_10_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_10_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_10_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self">Th61</a></i>;<br/><b>hence </b><a NAME="E3:67_1_10_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_10_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_10_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_10_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_10_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_10_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_10_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_10_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_10_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_10_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_10"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_10_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11"><b>suppose </b></a><a NAME="E1:67_1_11"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_11"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_11_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_11_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_11_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_11_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_11_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_11_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_11_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_11_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_11_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_11_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_11_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_11_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_11_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_11_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_11_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_11_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_11_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_11_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_11_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_11_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_11_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_11_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_11_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_11_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_11_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_11_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_11"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_11_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12"><b>suppose </b></a><a NAME="E1:67_1_12"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_12"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_12_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_12_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_12_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_12_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_12_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_12_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_12_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_12_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_12_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_12_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_12_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_12_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_12_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_12_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_12_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_12_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_12_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_12_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_12_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_12_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_12_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_12_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_12_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_12_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_12_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_12_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13"><b>suppose </b></a><a NAME="E1:67_1_13"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K2" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K2" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_13"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_13_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_13_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_13_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_13_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_13_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_13_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_13_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_13_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_13_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_13_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_13_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_13_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_13_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_13_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_13_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_13_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_13_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_13_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_13_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_13_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_13_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_13_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_13_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_13_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_13_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_13_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_13"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_13_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14"><b>suppose </b></a><a NAME="E1:67_1_14"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K1" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_14"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_14_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_14_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_14_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_14_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_14_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_14_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_14_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_14_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_14_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_14_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_14_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_14_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_14_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_14_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_14_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_14_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_2"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T61" target="_self">Th61</a></i>;<br/><b>hence </b><a NAME="E3:67_1_14_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_14_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_14_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_14_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_14_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_14_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_14_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_14_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_14_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_14"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_14_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15"><b>suppose </b></a><a NAME="E1:67_1_15"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_15"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_15_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_15_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_15_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_15_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_15_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_15_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_15_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_15_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_15_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_15_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_15_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_15_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_15_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_15_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_15_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_15_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_15_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_15_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_15_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_15_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_15_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_15_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_15_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_15_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_15_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_15"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_15_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16"><b>suppose </b></a><a NAME="E1:67_1_16"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="measure5.html#K3" target="_self">].</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:67_1_16"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_16_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_16_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_16_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1_16_1_1_1_1"/><i><font color="Green">E63</font></i>: 
(  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_16_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16_1_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_16_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:67_1_16_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_16_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:67_1_16_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:67_1_16_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E3:67_1_16_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:67_1_16_1_1_1"/><i><font color="Green">E63</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E4:67_1_16_1_1_1"/>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_16_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_1"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_16_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_16_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6" target="_self">{}</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:67_1_16_1_1_2"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:67_1_16_1_1_3"/><i><font color="Green">E63</font></i>: 
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="txt" href="measure5.html#E1:67_1_16_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:67_1_16_1_1_3"/>
( (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16_1_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:67_1_16_1_1_3"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure5.html#E2:67_1_16_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="measure5.html#E3:67_1_16_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="measure5.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_16_1_1_4"><b>suppose </b></a><a NAME="E1:67_1_16_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:67_1_16_1_1_4"/><b>then </b>
(  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure5.html#E1:67_1_16"><i><font color="Green">E60</font></i></a>, <a class="ref" href="measure5.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_16_1_1_4"/>
 <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure5.html#K5" target="_self">vol</a> <font color="Maroon">c<sub>2</sub></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>
