<?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="E1:48"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V3" target="_self">bounded_above</a>
 ;<br/>

<a NAME="E2:48"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
not <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#E1:48"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E3:48_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<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="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:48_1"/>
<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="48_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1"/><i><font color="Green">E37</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/>

<a NAME="E2:48_1_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#K2" target="_self">-infty</a> </span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="txt" href="supinf_1.html#E1:48_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E3:48_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:48_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="supinf_1.html#E1:48_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="supinf_1.html#E1:48_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E4:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T6">XXREAL_0:6</a></i>;<br/>
<b>hence </b><a NAME="E5:48_1_1_1"/>
<font color="Maroon">c<sub>3</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#K2" 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="E3:48_1_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#K2" 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/>
<a NAME="E4:48_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> </span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<b>hence </b><a NAME="E5:48_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:48"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#T64" target="_self">Th64</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_1"/>
<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#E3:48_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 ;<br/>


</div>
