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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<a NAME="E2:56"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:56_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>1</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>4</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>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</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#E1:56"><i><font color="Green">E50</font></i></a>, <a class="ref" href="measure5.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E4:56_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span>
 
<b>by </b><i><a class="txt" href="measure5.html#E3:56_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="measure5.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:56_1"/>
not <font color="Maroon">c<sub>5</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span>
 
<b>by </b><i><a class="txt" href="measure5.html#E3:56_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:56_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E4:56_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</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>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="measure5.html#E1:56"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1"><b>suppose </b></a><a NAME="E1:56_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E2:56"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2"><b>suppose </b></a><a NAME="E1:56_2_2"/><i><font color="Green">E52</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>1</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_2_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_2_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 <b>by </b><i><a class="txt" href="measure5.html#E2:56"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_2_2_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:56_2_2_1_1_2"/><i><font color="Green">E54</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>5</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>1</sub></font> &amp; <font color="Maroon">c<sub>5</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#E1:56_2_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="measure5.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:56_2_2_1_1_2"/>
( <font color="Maroon">c<sub>5</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Maroon">c<sub>5</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> )
 <b>by </b><i><a class="txt" href="measure5.html#E1:56_2_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="measure5.html#E3:56_2_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="measure5.html#D3" target="_self">Def3</a>, <a class="ref" href="measure5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E5:56_2_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:56_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( ( <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; not <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) or ( not <font color="Olive">b<sub>1</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>1</sub></font></span><a href="measure5.html#K3" target="_self">.]</a></span> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4" target="_self">.[</a></span> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
