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

<b>let </b><font color="Maroon">X</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> , <font color="Maroon">Y</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="E1:38"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">X</font> is <a href="supinf_1.html#V4">bounded_below</a> &amp; <font color="Maroon">Y</font> is <a href="supinf_1.html#V4">bounded_below</a> )
 ;<br/>

<a NAME="E2:38"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</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="38_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">LBx</font> being    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:38_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">LBx</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#E6:38_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_1.html#D12">SUPINF_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">LBy</font> being    <a href="supinf_1.html#M2">minorant</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:38_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">LBy</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#E6:38_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="supinf_1.html#D12">SUPINF_1:def 12</a></i>;<br/>
<a NAME="E5:38_1"/><i><font color="Green">E30</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:38_1_1"/><b>then </b>
<font color="Maroon">LBx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_2.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:38_1"/><i><font color="Green">E31</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:38_1_2"/><b>then </b>
<font color="Maroon">LBy</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_2.html#T4" target="_self">Th4</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:38_1"/><i><font color="Green">E50</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_3"/>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/>

<a NAME="E2:38_1_3"/><b>then </b>
<font color="Maroon">LBx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_2.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:38_1"/>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_4"/>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/>

<a NAME="E2:38_1_4"/><b>then </b>
<font color="Maroon">LBy</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_2.html#T4" target="_self">Th4</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <b>by </b><i><a class="ref" href="supinf_2.html#T5" target="_self">Th5</a>, <a class="ref" href="supinf_2.html#T6" target="_self">Th6</a>, <a class="ref" href="supinf_2.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
<span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">X</font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <font color="Maroon">Y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="supinf_2.html#K5" target="_self">+</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
