<?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> ;<br/>

<a NAME="E1:58"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:58"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K4" target="_self">SetMajorant</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:58"/>
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K4" target="_self">SetMajorant</a> <font color="Maroon">X</font> holds <br/> <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K4" target="_self">SetMajorant</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b>
<font color="Maroon">y</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:58_1"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:58"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of  <a href="supinf_1.html#K4" target="_self">SetMajorant</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:58"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E6:58"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E2:58"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E7:58"/>
for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:58_2"/><b>then </b>
<font color="Maroon">y</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E2:58"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:58_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:58"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:58"/>
(  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K4" target="_self">SetMajorant</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self">Def10</a>, <a class="ref" href="supinf_1.html#T36" target="_self">Th36</a>, <a class="ref" href="supinf_1.html#T39" target="_self">Th39</a>, , </i>;<br/>


</div>
