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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> &amp; <font color="Maroon" title="c2">Y</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> implies <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> &amp; <font color="Maroon" title="c2">Y</font> is <a href="supinf_1.html#V4" title="SUPINF_1:attr.4">bounded_below</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E2:40"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c3">LBx</font> being    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:40_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">LBx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">LBy</font> being    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E4:40_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c4">LBy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></i>;<br/>
<a NAME="E5:40_1"/><i><font color="Green" title="E34">A4</font></i>: 
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:40_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="supinf_2.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a>, <a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E6:40_1"/><i><font color="Green" title="E35">A5</font></i>: 
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E4:40_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="supinf_2.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a>, <a class="ref" href="supinf_1.html#D10" title="SUPINF_1:def.10">SUPINF_1:def 10</a></i>;<br/>
<a NAME="E7:40_1"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:40_1_1"/><b>then </b>
<font color="Maroon" title="c3">LBx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:40_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="supinf_2.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:40_1"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_2"/>
 <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:40_1_2"/><b>then </b>
<font color="Maroon" title="c4">LBy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" title="SUPINF_1:def.17">SUPINF_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:40_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="supinf_2.html#E8"><i><font color="Green" title="E6">Lm1</font></i></a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:40_1"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp;  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 <b>by </b><i><a class="txt" href="supinf_2.html#E5:40_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="supinf_2.html#E6:40_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="supinf_2.html#E7:40_1"><i><font color="Green" title="E36">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T27" target="_self" title="SUPINF_2:th.27">Th27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
