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

<b>let </b><font color="Maroon" title="c1">X</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#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E43">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="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">LB0</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="E3:60"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">LB0</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="ref" href="supinf_1.html#D12" target="_self" title="SUPINF_1:def.12">Def12</a></i>;<br/>
<a NAME="E4:60"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c2">LB0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:60"/><i><font color="Green" title="E46">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:60"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="supinf_1.html#T60" target="_self" title="SUPINF_1:th.60">Th60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E8:60"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>;<br/>
<a NAME="E9:60"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:60"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>
<a NAME="E10:60"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1: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#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="supinf_1.html#E25"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:60"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</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_1.html#E3:60"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="supinf_1.html#E4:60"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T10" title="XXREAL_0:th.10">XXREAL_0:10</a></i>;<br/>


</div>
