<?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  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="E1:37"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V3">bounded_above</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="supinf_1.html#V3">bounded_above</a> )
 ;<br/>

<a NAME="E2:37"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:37_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</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="supinf_2.html#E1:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_1.html#D11">SUPINF_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:37_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</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="supinf_2.html#E1:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_1.html#D11">SUPINF_1:def 11</a></i>;<br/>
<a NAME="E5:37_1"/><i><font color="Green">E34</font></i>: 
not  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1"/>
 <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:37_1_1"/><b>then </b>
 <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:37_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37_1"/><i><font color="Green">E35</font></i>: 
not  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_2"/>
 <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:37_1_2"/><b>then </b>
 <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:37_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:37_1"/><i><font color="Green">E36</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_3"/>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:37_1_3"/><b>then </b>
 <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:37_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:37_1"/>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_4"/>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:37_1_4"/><b>then </b>
 <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:37_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:37_1"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <b>by </b><i><a class="txt" href="supinf_2.html#E5:37_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="supinf_2.html#E6:37_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="supinf_2.html#E7:37_1"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
 <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K5" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T26" target="_self">Th26</a></i>;<br/>


</div>
