<?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> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/>

<a NAME="E2:32"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:32_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E1:32_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:32_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="txt" href="measure6.html#E1:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="measure6.html#E1:32_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="measure5.html#D2">MEASURE5:def 2</a></i>;<br/>
<a NAME="E5:32_1"/><b>then </b>
<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="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:32_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:32_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1">.]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E1:32_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:32_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="txt" href="measure6.html#E1:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="measure6.html#E1:32_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E5:32_2"/><b>then </b>
<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="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:32_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:32"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:32_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E1:32_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:32_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="txt" href="measure6.html#E1:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="measure6.html#E1:32_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="measure5.html#D4">MEASURE5:def 4</a></i>;<br/>
<a NAME="E5:32_3"/><b>then </b>
<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="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:32"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:32_4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3">.]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="measure6.html#E1:32_4"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="supinf_1.html#T10">SUPINF_1:10</a></i>;<br/>
<a NAME="E4:32_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="txt" href="measure6.html#E1:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="measure6.html#E1:32_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="measure5.html#D3">MEASURE5:def 3</a></i>;<br/>
<a NAME="E5:32_4"/><b>then </b>
<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="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>
<b>hence </b><a NAME="E6:32_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
( <span class="p1"><a href="measure5.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K2">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  &amp; <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  &amp; <span class="p1"><a href="measure5.html#K4">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K4">.[</a></span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  &amp; <span class="p1"><a href="measure5.html#K3">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="measure5.html#K3">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  )
 <b>by </b><i><a class="txt" href="measure6.html#E2:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="measure6.html#E3:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="measure6.html#E4:32"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>


</div>
