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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E2:39"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:39_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:39_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:39_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>
<a NAME="E5:39"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<a NAME="E1:39_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:39_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="supinf_1.html#D12" target="_self">Def12</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="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E4:39_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E7:39_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E4:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E8:39_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:39_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_1.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:39_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E4:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E4:39"><i><font color="Green">E26</font></i></a>, <a class="txt" href="supinf_1.html#E1:39_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E7:39"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 <b>by </b><i><a class="txt" href="supinf_1.html#E4:39"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
